Seven deadly sins of set theory

[WM]

1. Scrooge McDuck's bankrupt

Scrooge Mc Duck earns 1000 $ daily and spends only 1 $ per day. As a
cartoon-figure he will live forever and his wealth will increase without
bound. But according to set theory he will get bankrupt if he spends the
dollars in the same order as he receives them. Only if he always spends
them in another order, for instance every day the second dollar
received, he will get rich. These different results prove set theory to
be useless for all practical purposes.

The above story is only the story of Tristram Shandy in simplified
terms, which has been narrated by Fraenkel, one of the fathers of ZF set
theory.

"Well known is the story of Tristram Shandy who undertakes to write his
biography, in fact so pedantically, that the description of each day
takes him a full year. Of course he will never get ready if continuing
that way. But if he lived infinitely long (for instance a 'countable
infinity' of years [...]), then his biography would get 'ready',
because, expressed more precisely, every day of his life, how late ever,
finally would get its description because the year scheduled for this
work would some time appear in his life." [A. Fraenkel: "Einleitung in
die Mengenlehre", 3rd ed., Springer, Berlin (1928) p. 24] "If he is
mortal he can never terminate; but did he live forever then no part of
his biography would remain unwritten, for to each day of his life a year
devoted to that day's description would correspond." [A.A. Fraenkel, A.
Levy: "Abstract set theory", 4th ed., North Holland, Amsterdam (1976) p.
30]

2. Failed enumeration of the fractions

All natural numbers are said to be enough to index all positive
fractions. This can be disproved when the natural numbers are taken from
the first column of the matrix of all positive fractions

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
.. .

To cover the whole matrix by the integer fractions amounts to the idea
that the letters X in

XOOO...
XOOO...
XOOO...
XOOO...
..

can be redistributed to cover all positions by exchanging them with the
letters O. (X and O must be exchanged because where an index has left,
there is no index remaining.) But where should the O remain if not
within the matrix at positions not covered by X?

3. Violation of translation invariance

Translation invariance is fundamental to every scientific theory. With n
m ∈ ℕ and q ∈ {ℚ ∩ (0, 1]} there is precisely the same number of
rational points n + q in (n, n+1] as of rational points m + q in (m,
m+1] . However, half of all positive rational numbers of Cantor's
enumeration
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2,
5/1, ...
are of the form 0 + q and lie in the first unit interval between 0 and
1. There are less rational points in (1, 2] but more than in (2, 3] and
so on.

4. Violation of inclusion monotony

Every endsegment E(n) = {n, n+1, n+2, ...} of natural numbers has an
infinite intersection with all other infinite endsegments.
∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀ .
Set theory however comes to the conclusion that there are only infinite
endsegments and that their intersection is empty. This violates the
inclusion monotony of the endegments according to which, as long as only
non-empty endsegments are concerned, their intersection is non-empty.

5. Actual infinity implies a smallest unit fraction

All unit fractions 1/n have finite distances from each other
∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0.
Therefore the function Number of Unit Fractions between 0 and x, NUF(x),
cannot be infinite for all x > 0. The claim of set theory
∀x ∈ (0, 1]: NUF(x) = ℵo
is wrong. If every positive point has ℵo unit fractions at its left-hand
side, then there is no positive point with less than ℵo unit fractions
at its left-hand side, then all positive points have ℵo unit fractions
at their left-hand side, then the interval (0, 1] has ℵo unit fractions
at its left-hand side, then ℵo unit fractions are negative.
Contradiction.

6. There are more path than nodes in the infinite Binary Tree

Since each of n paths in the complete infinite Binary Tree contains at
least one node differing from all other paths, there are not less nodes
than paths possible. Everything else would amount to having more houses
than bricks.

7. The diagonal does not define a number

An endless digit sequence without finite definition of the digits cannot
define a real number. After every known digit almost all digits will
follow.

Regards, WM

[Ross Finlayson]

Post by WM
1. Scrooge McDuck's bankrupt
Scrooge Mc Duck earns 1000 $ daily and spends only 1 $ per day. As a
cartoon-figure he will live forever and his wealth will increase without
bound. But according to set theory he will get bankrupt if he spends the
dollars in the same order as he receives them. Only if he always spends
them in another order, for instance every day the second dollar
received, he will get rich. These different results prove set theory to
be useless for all practical purposes.
The above story is only the story of Tristram Shandy in simplified
terms, which has been narrated by Fraenkel, one of the fathers of ZF set
theory.
"Well known is the story of Tristram Shandy who undertakes to write his
biography, in fact so pedantically, that the description of each day
takes him a full year. Of course he will never get ready if continuing
that way. But if he lived infinitely long (for instance a 'countable
infinity' of years [...]), then his biography would get 'ready',
because, expressed more precisely, every day of his life, how late ever,
finally would get its description because the year scheduled for this
work would some time appear in his life." [A. Fraenkel: "Einleitung in
die Mengenlehre", 3rd ed., Springer, Berlin (1928) p. 24] "If he is
mortal he can never terminate; but did he live forever then no part of
his biography would remain unwritten, for to each day of his life a year
devoted to that day's description would correspond." [A.A. Fraenkel, A.
Levy: "Abstract set theory", 4th ed., North Holland, Amsterdam (1976) p.
30]
2. Failed enumeration of the fractions
All natural numbers are said to be enough to index all positive
fractions. This can be disproved when the natural numbers are taken from
the first column of the matrix of all positive fractions
1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
.. .
To cover the whole matrix by the integer fractions amounts to the idea
that the letters X in
XOOO...
XOOO...
XOOO...
XOOO...
..
can be redistributed to cover all positions by exchanging them with the
letters O. (X and O must be exchanged because where an index has left,
there is no index remaining.) But where should the O remain if not
within the matrix at positions not covered by X?
3. Violation of translation invariance
Translation invariance is fundamental to every scientific theory. With n
m ∈ ℕ and q ∈ {ℚ ∩ (0, 1]} there is precisely the same number of
rational points n + q in (n, n+1] as of rational points m + q in (m,
m+1] . However, half of all positive rational numbers of Cantor's
enumeration
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2,
5/1, ...
are of the form 0 + q and lie in the first unit interval between 0 and
1. There are less rational points in (1, 2] but more than in (2, 3] and
so on.
4. Violation of inclusion monotony
Every endsegment E(n) = {n, n+1, n+2, ...} of natural numbers has an
infinite intersection with all other infinite endsegments.
∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀ .
Set theory however comes to the conclusion that there are only infinite
endsegments and that their intersection is empty. This violates the
inclusion monotony of the endegments according to which, as long as only
non-empty endsegments are concerned, their intersection is non-empty.
5. Actual infinity implies a smallest unit fraction
All unit fractions 1/n have finite distances from each other
∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0.
Therefore the function Number of Unit Fractions between 0 and x, NUF(x),
cannot be infinite for all x > 0. The claim of set theory
∀x ∈ (0, 1]: NUF(x) = ℵo
is wrong. If every positive point has ℵo unit fractions at its left-hand
side, then there is no positive point with less than ℵo unit fractions
at its left-hand side, then all positive points have ℵo unit fractions
at their left-hand side, then the interval (0, 1] has ℵo unit fractions
at its left-hand side, then ℵo unit fractions are negative.
Contradiction.
6. There are more path than nodes in the infinite Binary Tree
Since each of n paths in the complete infinite Binary Tree contains at
least one node differing from all other paths, there are not less nodes
than paths possible. Everything else would amount to having more houses
than bricks.
7. The diagonal does not define a number
An endless digit sequence without finite definition of the digits cannot
define a real number. After every known digit almost all digits will
follow.
Regards, WM

I replied to this in the other identical thread now you're spamming in this one.

Your problems include set theory has problems, but you're not helping.

Really if you want to address set theory's problems then you'll review my slates,
about uncountability and logical paradox, about continuous domains and foundations.

So, "Mister McDuck", or, "Mister Magoo", as it were, first you should read "Hodges'
Hopeless: an editor recalls some papers", about usual mistakes in studying set theory,
then strike all those examples from yours, then what you're left with is to read mine.

...
way it is

[immibis]

[snip]

what point are you trying to make? infinity is strange

[WM]

Post by immibis
what point are you trying to make? infinity is strange

But it is based on logic. This logic is violated in the seven points I
made.

Regards, WM

[Richard Damon]

Post by WM

Post by immibis
what point are you trying to make? infinity is strange

But it is based on logic. This logic is violated in the seven points I
made.
Regards, WM

Which are base on logic that doesn't handle infinity.

Yes, it is well know that infinite sets break some of the seemingly
obvious properties that hold for finite sets.

YOU just don't seem to understand and accept that fact, and keep on
making the ERROR of asuming it must.

[WM]

Post by Richard Damon

Post by WM

Post by immibis
what point are you trying to make? infinity is strange

But it is based on logic. This logic is violated in the seven points I
made.

Which are base on logic that doesn't handle infinity.

This logic is indispensable.

Post by Richard Damon
Yes, it is well know that infinite sets break some of the seemingly
obvious properties that hold for finite sets.

But it has not yet been recognized that ZF breaks indispensable laws of
logic.

Post by Richard Damon
YOU just don't seem to understand and accept that fact, and keep on
making the ERROR of asuming it must.

That is not an error. It shows that ZF could never acquire any relevance
for reality where the basic laws of logic are referenced. It shows that ZF
is only a religion to be believed by its proponents and poor captured
students.

Regards, WM

[Richard Damon]

Post by WM

Post by Richard Damon

Post by WM

Post by immibis
what point are you trying to make? infinity is strange

But it is based on logic. This logic is violated in the seven points
I made.

Which are base on logic that doesn't handle infinity.

This logic is indispensable.

Maybe for your small mind.

Post by WM

Post by Richard Damon
Yes, it is well know that infinite sets break some of the seemingly
obvious properties that hold for finite sets.

But it has not yet been recognized that ZF breaks indispensable laws of
logic.

No, it shows that some supposed laws of logic can't handle unbounded sets.

Post by WM

Post by Richard Damon
YOU just don't seem to understand and accept that fact, and keep on
making the ERROR of asuming it must.

That is not an error. It shows that ZF could never acquire any relevance
for reality where the basic laws of logic are referenced. It shows that
ZF is only a religion to be believed by its proponents and poor captured
students.
Regards, WM

In other words, you don't understand what it says so you ignore it.

[Jim Burns]

Post by WM
7. The diagonal does not define a number
An endless digit sequence without
finite definition of the digits
cannot define a real number.
After every known digit
almost all digits will follow.

After every known and unknown digit,
almost all digits follow.

For each digit in the sequence,
the cardinal of digits.before is ⁺¹.able,
the cardinal of digits.after is
larger than each ⁺¹.able cardinal,
and thus is not.⁺¹.able.


It is sufficient that
an endless digit sequence without
finite definition of the digits
exist.

There is at most one real number
which is permitted by each
finite initial sub.sequence of digits.
That one real number is the one which
the endless digit sequence represents.

| Assume otherwise.
| Assume two points at a distance d > 0
| are permitted by each
| finite initial sub.sequence of digits.
|
| However,
| there is an initial n.digit sub.sequence which
| permits only points apart by 10-n < d
| Those two points cannot both be permitted
| by that n.digit sub.sequence or
| by each finite initial sub.sequence.
| Contradiction.

Therefore,
there is at most one real number
which is permitted by each
finite initial sub.sequence of digits.

Also,
there is at least one real number
which is permitted by each
finite initial sub.sequence of digits.

That follows from the requirement that
functions which jump
are discontinuous at some point.

For each endless digit sequence,
each digit preceded by ⁺¹.ably.many digits,
there is no more and no less than
one point.

For each point,
there is no less than one and
no more than one
(no more than two for trailing 0's and 9's)
endless digit sequence,
each digit preceded by ⁺¹.ably.many digits,

The representation of real points by digits
is obviously cousin to
the representation of rational points by digits.

However, the cousins are not the same.
The points _exist_
The digit.sequences _exist_
We know that by augmenting descriptive claims
with only not.first.false claims.
That isn't a calculation in the sense that
a rational point is calculated.

Nevertheless, we know they exist, uncalculated,
because we know that
a finite sequence of _claims_
if it has no first.false _claim_
has no false _claim_

[Ross Finlayson]

Post by Jim Burns

Post by WM
7. The diagonal does not define a number
An endless digit sequence without
finite definition of the digits
cannot define a real number.
After every known digit
almost all digits will follow.

After every known and unknown digit,
almost all digits follow.
For each digit in the sequence,
the cardinal of digits.before is ⁺¹.able,
the cardinal of digits.after is
larger than each ⁺¹.able cardinal,
and thus is not.⁺¹.able.
It is sufficient that
an endless digit sequence without
finite definition of the digits
exist.
There is at most one real number
which is permitted by each
finite initial sub.sequence of digits.
That one real number is the one which
the endless digit sequence represents.
| Assume otherwise.
| Assume two points at a distance d > 0
| are permitted by each
| finite initial sub.sequence of digits.
|
| However,
| there is an initial n.digit sub.sequence which
| permits only points apart by 10-n < d
| Those two points cannot both be permitted
| by that n.digit sub.sequence or
| by each finite initial sub.sequence.
| Contradiction.
Therefore,
there is at most one real number
which is permitted by each
finite initial sub.sequence of digits.
Also,
there is at least one real number
which is permitted by each
finite initial sub.sequence of digits.
That follows from the requirement that
functions which jump
are discontinuous at some point.
For each endless digit sequence,
each digit preceded by ⁺¹.ably.many digits,
there is no more and no less than
one point.
For each point,
there is no less than one and
no more than one
(no more than two for trailing 0's and 9's)
endless digit sequence,
each digit preceded by ⁺¹.ably.many digits,
The representation of real points by digits
is obviously cousin to
the representation of rational points by digits.
However, the cousins are not the same.
The points _exist_
The digit.sequences _exist_
We know that by augmenting descriptive claims
with only not.first.false claims.
That isn't a calculation in the sense that
a rational point is calculated.
Nevertheless, we know they exist, uncalculated,
because we know that
a finite sequence of _claims_
if it has no first.false _claim_
has no false _claim_

If it's sufficient to establish a model of arithmetic then
by the GIT's you'll agree it's at best incomplete.

... That it's false to say it's, the "true", claim.
Only scientific and not falsified.

Platonism then sort of demands "there are true
numbers, so work it up".

"There is no 'but', only 'yet', ...."

...

[Jim Burns]

On Friday, January 5, 2024

Post by Jim Burns

Post by WM
An endless digit sequence without
finite definition of the digits
cannot define a real number.
After every known digit
almost all digits will follow.

The representation of real points by digits
is obviously cousin to
the representation of rational points by digits.
However, the cousins are not the same.
The points _exist_
The digit.sequences _exist_
We know that by augmenting descriptive claims
with only not.first.false claims.
That isn't a calculation in the sense that
a rational point is calculated.
Nevertheless, we know they exist, uncalculated,
because we know that
a finite sequence of _claims_
if it has no first.false _claim_
has no false _claim_

If it's sufficient
to establish a model of arithmetic
then by the GIT's
you'll agree it's at best incomplete.

I'm not sure we're on the same page
with regard to incompleteness.

Incomplete == some things we can't know.
Were you (RF) seriously contemplating
not (some things we can't know)?

Gödel didn't surprise with the result itself.
Yes, some things we can't know. Whatever.
Gödel surprised with _proving_ it.
| On Formally Undecidable Propositions of
| Principia Mathematica and Related Systems
|
_Formally_ undecidable.

----
If a theory has a model,
then it's consistent.

If a theory can express
recursive definitions
non.recursively
(re.stated without the defined term)
(which is something arithmetic can do),
then
the theory can _quote_
recursive definitions of
what it is to be a formula and
what it is to be a proof.

By "quote", I mean: represent
an object of language (formula, proof) by
an object of study (number, set)
(Gödel.numbers et al)

If a theory can quote
what it is to be a formula and
what it is to be a proof,
then the theory can express G(x) such that
G("H(x)") ⟹ proof of H("H(x)") not.exists.
¬G("H(x)") ⟹ proof of contradiction exists.

If a theory can express G("H(x)")
then the theory can express G("G(x)")
G("G(x)") ⟹ proof of G("G(x)") not.exists.
¬G("G(x)") ⟹ proof of contradiction exists.

If a theory has a certain
very.attainable level of expressiveness,
then choose one: incomplete or inconsistent.

But there is a model. It's consistent.
So, it's incomplete.

... That it's false to say
it's, the "true", claim.
Only scientific and not falsified.
Platonism then sort of demands
"there are true numbers, so work it up".
"There is no 'but', only 'yet', ...."

We know that
each theory is true of
whatever that theory is true of.

Some theories are true of nothing.
Contradictions can be proved in those.

Some theories are true of something,
of what we intend or of something unintended.
Contradictions cannot be proved in those.

Some theories, numbers, for example,
if they can prove they're consistent,
would prove that they _aren't_ consistent.

There is no "yet" to not.proving consistency.
We know there is no such proof --
or, if (sadly) there is, it's meaningless,
about nothing.

However,
there are better options than
a theory proving itself consistent.

Gödel's work does not deny that
some other theory can prove
our theory consistent.

For example,
ZFC is a theorem of
ZFC+inaccessible.cardinal.exists

Some theories are so simple that,
even though we know there is no proof,
we don't take seriously the possibility
that they have no model.

For example,
the set.theory fragment,
-- {} exists
-- x∪{y} exists
-- same.element.sets x and y are equal
proves arithmetic and Gödel.incompleteness

[Ross Finlayson]

Post by Jim Burns

On Friday, January 5, 2024

Post by Jim Burns

Post by WM
An endless digit sequence without
finite definition of the digits
cannot define a real number.
After every known digit
almost all digits will follow.

The representation of real points by digits
is obviously cousin to
the representation of rational points by digits.
However, the cousins are not the same.
The points _exist_
The digit.sequences _exist_
We know that by augmenting descriptive claims
with only not.first.false claims.
That isn't a calculation in the sense that
a rational point is calculated.
Nevertheless, we know they exist, uncalculated,
because we know that
a finite sequence of _claims_
if it has no first.false _claim_
has no false _claim_

If it's sufficient
to establish a model of arithmetic
then by the GIT's
you'll agree it's at best incomplete.

I'm not sure we're on the same page
with regard to incompleteness.
Incomplete == some things we can't know.
Were you (RF) seriously contemplating
not (some things we can't know)?
Gödel didn't surprise with the result itself.
Yes, some things we can't know. Whatever.
Gödel surprised with _proving_ it.
| On Formally Undecidable Propositions of
| Principia Mathematica and Related Systems
|
_Formally_ undecidable.
----
If a theory has a model,
then it's consistent.
If a theory can express
recursive definitions
non.recursively
(re.stated without the defined term)
(which is something arithmetic can do),
then
the theory can _quote_
recursive definitions of
what it is to be a formula and
what it is to be a proof.
By "quote", I mean: represent
an object of language (formula, proof) by
an object of study (number, set)
(Gödel.numbers et al)
If a theory can quote
what it is to be a formula and
what it is to be a proof,
then the theory can express G(x) such that
G("H(x)") ⟹ proof of H("H(x)") not.exists.
¬G("H(x)") ⟹ proof of contradiction exists.
If a theory can express G("H(x)")
then the theory can express G("G(x)")
G("G(x)") ⟹ proof of G("G(x)") not.exists.
¬G("G(x)") ⟹ proof of contradiction exists.
If a theory has a certain
very.attainable level of expressiveness,
then choose one: incomplete or inconsistent.
But there is a model. It's consistent.
So, it's incomplete.

... That it's false to say
it's, the "true", claim.
Only scientific and not falsified.
Platonism then sort of demands
"there are true numbers, so work it up".
"There is no 'but', only 'yet', ...."

We know that
each theory is true of
whatever that theory is true of.
Some theories are true of nothing.
Contradictions can be proved in those.
Some theories are true of something,
of what we intend or of something unintended.
Contradictions cannot be proved in those.
Some theories, numbers, for example,
if they can prove they're consistent,
would prove that they _aren't_ consistent.
There is no "yet" to not.proving consistency.
We know there is no such proof --
or, if (sadly) there is, it's meaningless,
about nothing.
However,
there are better options than
a theory proving itself consistent.
Gödel's work does not deny that
some other theory can prove
our theory consistent.
For example,
ZFC is a theorem of
ZFC+inaccessible.cardinal.exists
Some theories are so simple that,
even though we know there is no proof,
we don't take seriously the possibility
that they have no model.
For example,
the set.theory fragment,
-- {} exists
-- x∪{y} exists
-- same.element.sets x and y are equal
proves arithmetic and Gödel.incompleteness

It's not so much "Goedel's Incompleteness Theorems knows unknowables",
as, "Goedel's Incompleteness Theorems show there are more truths than
what are already theorems". There's, "the extra-ordinary".

When you say model, that's a structure, it's structuralism, and for constructivists.

When you say "some theories are true of nothing, contradictions can be proved in them",
that's really a great statement for Ex Falso Nihilum contra Ex Falso Quodlibet,
and it's true and it's good.

So, consider the theories of "cardinals in ordinals", and, "ordinals in cardinals", as
two theories about one theory, with different primary objects, elementarily.
This is already framed as "ordering theory" vis-a-vis "set theory". (Then,
the goal is that it's one theory, or that the one theory has structure the
objects either way.)

So, Goedel's Incompleteness Theorems, have their sorts of beginnings and ends.

I.e., a plain sequence of Geodel functions still carries on out to infinity.

As a complement to Goedel's, don't forget to include Cohen's forcing,
he tops it off with an ordinal its own order type. ("Extra-ordinary.")

It's not so much so that there are large cardinals, which aren't cardinals nor sets,
in set theory. It's usually then about class/set distinction, but, it results that
it's better to have the extra-ordinary theory up front then find set theory in that.

Then, this is related to my slates of uncountability and logical paradox,
about DesCartes to Cohen, a sort of paleo-classical super-modern Platonism.

"In my 10,000's posts to sci.math, sci.logic, sci.physics, sci.physics.relativity, ...."

Good luck dear sir and thanks for your reply, what you can find is that "A Theory"
and the "Null Axiom Theory" really represent A Theory.


...

[WM]

Post by Jim Burns

Post by WM
7. The diagonal does not define a number
An endless digit sequence without
finite definition of the digits
cannot define a real number.
After every known digit
almost all digits will follow.

It is sufficient that
an endless digit sequence without
finite definition of the digits
exist.

No. Even and endless digit sequence does not describe an irrational
number. The irrational number is the limit only. Compare 0.999... which
does not contain 1 but only has the limit 1.

Post by Jim Burns
There is at most one real number
which is permitted by each
finite initial sub.sequence of digits.

There are infinitely many. Therefore it is impossible to decide which real
number is described.

Post by Jim Burns
Therefore,
there is at most one real number
which is permitted by each
finite initial sub.sequence of digits.

By each defined digit sequence infinitely many numbers are permitted.
And the dark digits cannot describe any real number.

Regards, WM

[Jim Burns]

Post by WM

Post by Jim Burns

Post by WM
An endless digit sequence without
finite definition of the digits
cannot define a real number.
After every known digit
almost all digits will follow.

There is at most one real number
which is permitted by each
finite initial sub.sequence of digits.

There are infinitely many.

Assume there are two permitted points.
Each power 10⁻ⁿ is larger than their distance,
a positive distance d > 0 apart.

β ≥ d > 0 is the least upper bound of
distances which each 10⁻ⁿ is larger than.

10β > β is not
a distance which each 10⁻ⁿ is larger than.
Thus b exists: 10⁻ᵇ < 10β

β/10 < β is
a distance which each 10⁻ⁿ is larger than.
Thus, in particular,
β/10 < 10⁻ᵇ⁻²

However,
(10⁻ᵇ)/100 < (10β)/100
10⁻ᵇ⁻² < β/10
Contradiction.

Therefore,
there _aren't_ two points which
are permitted by each
finite initial sub.sequence of
the endless digit sequence.

Post by WM
Therefore it is impossible to decide which
real number is described.

Whether or not we describe these points,
these points _exist_
no more than one point to each
endless digit sequence.

[WM]

Post by Jim Burns

Post by WM
Therefore it is impossible to decide which
real number is described.

Whether or not we describe these points,
these points _exist_
no more than one point to each
endless digit sequence.

Dark numbers, dark points and dark parts of infinite sequences exist. But
we cannot distinguish and use them. And we can prove that there are not
more irrational numbers than rational numbers. Just today I showed some
proofs to my students. One is this: Between ***every*** pair of irrational
numbers there is a rational number. Another one, which was very well
received, is the game Conquer the Binay Tree:

You start with one cent, buy a path in the Binary Tree and get one cent
for every covered node. Then you buy another path and get one cent for
every node not yet covered by the first path. You will never earn less
than one cent, because every path is distinct by at least one node from
every other path. Therefore you will not get bankrupt. But if there were
more paths than nodes, you would get bankrupt. Hence Cantor is defeated.

Regards, WM

[Ross Finlayson]

Post by WM

Post by Jim Burns

Post by WM
Therefore it is impossible to decide which
real number is described.

Whether or not we describe these points,
these points _exist_
no more than one point to each
endless digit sequence.

Dark numbers, dark points and dark parts of infinite sequences exist. But
we cannot distinguish and use them. And we can prove that there are not
more irrational numbers than rational numbers. Just today I showed some
proofs to my students. One is this: Between ***every*** pair of irrational
numbers there is a rational number. Another one, which was very well
You start with one cent, buy a path in the Binary Tree and get one cent
for every covered node. Then you buy another path and get one cent for
every node not yet covered by the first path. You will never earn less
than one cent, because every path is distinct by at least one node from
every other path. Therefore you will not get bankrupt. But if there were
more paths than nodes, you would get bankrupt. Hence Cantor is defeated.
Regards, WM

That's better than usual, and terms that aren't so mangled,
where sometimes "mangle" means flatten like a steam press
and other times means "spindle" as the over and under in definition,
so there is that rationals are HUGE like Friedman's,
and, there is a constructible tree like Hausdorff,
then Cantor is not "defeated", rather, refined in apologetics.

There is Cantor space, and, there is square Cantor space.

It's all the 0's and 1's, ....

....

[Jim Burns]

Post by WM

Post by Jim Burns

Post by WM

Post by Jim Burns

Post by WM
An endless digit sequence without
finite definition of the digits
cannot define a real number.
After every known digit
almost all digits will follow.

There is at most one real number
which is permitted by each
finite initial sub.sequence of digits.

There are infinitely many.
Therefore it is impossible to decide
which real number is described.

Whether or not we describe these points,
these points _exist_
no more than one point to each
endless digit sequence.

Dark numbers, dark points
and dark parts of infinite sequences
exist.

Elsewhere, you have told us that
darkᵂᴹ numbers and their ilk
are never one step away from
visibleᵂᴹ numbers and their ilk.

I'll clarify.
There is at most one point
which is permitted by each
finite initial sub.sequence of
visibleᵂᴹ digits.

Because,
if there are two permitted points,
there is a positive least upper bound β of
distances < each visibleᵂᴹ 10⁻ⁿ
β/10 < β < 10β
and visibleᵂᴹ 10⁻ᵇ < 10β
and visibleᵂᴹ 10⁻ᵇ⁻² > β/10
but also
10⁻ᵇ/100 < 10β/100
10⁻ᵇ⁻² < β/10
Contradiction.
Thus,
not two.

Post by WM
Just today I showed some proofs to my students.

If you are claiming your students for
some form of peer.review, then
you are accepting pre-calculus students as
_peers_

I won't object if you choose to do that.
I just want to make sure you understand that
they are who you are accepting as peers.

[Ross Finlayson]

Post by Jim Burns

Post by WM

Post by Jim Burns

Post by WM

Post by Jim Burns

Post by WM
An endless digit sequence without
finite definition of the digits
cannot define a real number.
After every known digit
almost all digits will follow.

There is at most one real number
which is permitted by each
finite initial sub.sequence of digits.

There are infinitely many.
Therefore it is impossible to decide
which real number is described.

Whether or not we describe these points,
these points _exist_
no more than one point to each
endless digit sequence.

Dark numbers, dark points
and dark parts of infinite sequences
exist.

Elsewhere, you have told us that
darkᵂᴹ numbers and their ilk
are never one step away from
visibleᵂᴹ numbers and their ilk.
I'll clarify.
There is at most one point
which is permitted by each
finite initial sub.sequence of
visibleᵂᴹ digits.
Because,
if there are two permitted points,
there is a positive least upper bound β of
distances < each visibleᵂᴹ 10⁻ⁿ
β/10 < β < 10β
and visibleᵂᴹ 10⁻ᵇ < 10β
and visibleᵂᴹ 10⁻ᵇ⁻² > β/10
but also
10⁻ᵇ/100 < 10β/100
10⁻ᵇ⁻² < β/10
Contradiction.
Thus,
not two.

Post by WM
Just today I showed some proofs to my students.

If you are claiming your students for
some form of peer.review, then
you are accepting pre-calculus students as
_peers_
I won't object if you choose to do that.
I just want to make sure you understand that
they are who you are accepting as peers.

Everybody knows the entire point of pre-calculus
is to make it clear that the usual notion of an Aristotelean
continuum, which is about the most usual sort of intuition
about the continuum after constant motion, and the
course-of-passage as through points in a line, is gently
shushing that down for students who do have such
an intuitive notion of analytical character, and then
explaining for all that the usual laws of arithmetic and
delta-epsilonics, together, make for defining infinite limit.

That is all.

--
Today we call it line-reals and don't use quasi-modal logic
And it's standard, too

[Jim Burns]

On Tuesday, January 9, 2024

Post by Jim Burns

Post by WM
Just today I showed some proofs to my students.

If you are claiming your students for
some form of peer.review, then
you are accepting pre-calculus students as
_peers_

Everybody knows
the entire point of pre-calculus is
to make it clear that the usual notion of
an Aristotelean continuum, [...]

I could have said that better.

In that post,
the only thing I meant by "pre-calculus"
was "having not yet been a calculus student".
I wasn't thinking of "being a student in
the course just before calculus".

Calculus and WM's dark.number game
make a poor fit together.
I expect WM to agree with that much,
even if he and I assign blame for
the poor fit differently.

I haven't actually heard from anyone that
WM's students are not.yet.calculus.
It only seems to me as though they must be.

Sorry for any confusion.

Everybody knows
the entire point of pre-calculus is
to make it clear that
the usual notion of
an Aristotelean continuum,
which is about
the most usual sort of intuition about
the continuum after constant motion,
and
the course-of-passage as through
points in a line,
is
gently shushing that down for
students who do have
such an intuitive notion of
analytical character,
and then
explaining for all
that the usual laws of
arithmetic and delta-epsilonics,
together,
make for defining infinite limit.

Gently shushing down notions which are
not the preferred notions
is the purpose for which we hold classes.

If it can be done,
I think that
giving the reasons that
some other notion is preferred
can be an excellent strategy for
effective gentle.shushing.down.

However,
in some instances,
we are, today, looking at preferences
resulting from resolution of a long controversy
between brilliant people
encyclopedically educated in their field.

Should we ask students,
upon their first contact with that field,
to reproduce
the best of that field?

If that is a reasonable ask,
that would be wonderful to see.

However,
it might well not.be a reasonable ask.

It might well be necessary for
students to _trust_ that
reasons exist for preferring
one set of notions over
another,
to trust, at least,
until they acquire their own
brilliance and
encyclopedic knowledge.

My own preference is
to share the beauty in
these deep notions,
with all and sundry,
with those with only
a momentary interest.

However,
I have to recognize that
I can't always get what I want.

[WM]

Post by Jim Burns
Calculus and WM's dark.number game
make a poor fit together.

Of course. Calculus uses potential infinity only. Dark numbers can exist
only in actual infinity, the complement of potential infinity.
Feferman's book In the light of logic answers the question: Is Cantor
necessary for the maths of the real world? with a resounding no.

Regards, WM

[Richard Damon]

Post by WM

Post by Jim Burns
Calculus and WM's dark.number game
make a poor fit together.

Of course. Calculus uses potential infinity only. Dark numbers can exist
only in actual infinity, the complement of potential infinity.
Feferman's book In the light of logic answers the question: Is Cantor
necessary for the maths of the real world? with a resounding no.
Regards, WM

But you claim them to be part of the sets that are only "Potential
Infinity", or are you retracting that claim. (All Natural Numbers are
only by your definition "Potentially Infinite")

All members of the Natural Numbers are Finite, so none of them are
"Actually Infinite".

The set of them has a SIZE that is infinite, but none of the members of
it ever are.

[Ross Finlayson]

Post by Richard Damon

Post by WM

Post by Jim Burns
Calculus and WM's dark.number game
make a poor fit together.

Of course. Calculus uses potential infinity only. Dark numbers can exist
only in actual infinity, the complement of potential infinity.
Feferman's book In the light of logic answers the question: Is Cantor
necessary for the maths of the real world? with a resounding no.
Regards, WM

But you claim them to be part of the sets that are only "Potential
Infinity", or are you retracting that claim. (All Natural Numbers are
only by your definition "Potentially Infinite")
All members of the Natural Numbers are Finite, so none of them are
"Actually Infinite".
The set of them has a SIZE that is infinite, but none of the members of
it ever are.

The order type of ordinals is an ordinal.

Finite ordinals as the classes of each those not containing themselves,
according to simple quantification or "the set of all sets that don't
contain themselves", contains itself.

Such are simple reasons why the infinite integers have an infinite integer.

Yes I know that ZF two axioms of restriction of comprehension or "don't look"
include one that is "there's an infinity that isn't thusly various".

So, you might find it interesting that in more naive theories,
it's those one-line proofs as above that do give that there are
true infinites courtesy the unbounded its quantification.


Then about MW's "I invoke the Absolute!", it's he's drunk
then also that potential and actual infinities do have differences
in real mathematics, one recalls "A Sober Mind Speaks".

So, RD's "there is only one theory and it ignores all my problems"
and MWs "there are at least two theories and they don't agree",
seems for a sort of "yet somehow, there's a theory".

... With laws, plural, of large numbers.

If you want a gentle introduction to set theory,
try reading Quine's Set Theory then after it introduces
the class-set distinction and x not in x and x not equals x,
and why neither of those is first-order in ZF, quit.


... in a theory, ....

[Richard Damon]

Post by Ross Finlayson

Post by Richard Damon

Post by WM

Post by Jim Burns
Calculus and WM's dark.number game
make a poor fit together.

Of course. Calculus uses potential infinity only. Dark numbers can exist
only in actual infinity, the complement of potential infinity.
Feferman's book In the light of logic answers the question: Is Cantor
necessary for the maths of the real world? with a resounding no.
Regards, WM

But you claim them to be part of the sets that are only "Potential
Infinity", or are you retracting that claim. (All Natural Numbers are
only by your definition "Potentially Infinite")
All members of the Natural Numbers are Finite, so none of them are
"Actually Infinite".
The set of them has a SIZE that is infinite, but none of the members of
it ever are.

The order type of ordinals is an ordinal.
Finite ordinals as the classes of each those not containing themselves,
according to simple quantification or "the set of all sets that don't
contain themselves", contains itself.
Such are simple reasons why the infinite integers have an infinite integer.
Yes I know that ZF two axioms of restriction of comprehension or "don't look"
include one that is "there's an infinity that isn't thusly various".
So, you might find it interesting that in more naive theories,
it's those one-line proofs as above that do give that there are
true infinites courtesy the unbounded its quantification.
Then about MW's "I invoke the Absolute!", it's he's drunk
then also that potential and actual infinities do have differences
in real mathematics, one recalls "A Sober Mind Speaks".
So, RD's "there is only one theory and it ignores all my problems"
and MWs "there are at least two theories and they don't agree",
seems for a sort of "yet somehow, there's a theory".
... With laws, plural, of large numbers.
If you want a gentle introduction to set theory,
try reading Quine's Set Theory then after it introduces
the class-set distinction and x not in x and x not equals x,
and why neither of those is first-order in ZF, quit.
... in a theory, ....

I understand the multitude of different ways to define "Numbers", but
their are a core set of fundamental numbers with basically agreed on
definitions, and reasonable people don't try calling something that
doesn't match the accepted definition as being one of those sets.

"Natural Numbers" is such a set, and while there may be several ways of
expressing its "generation", they all are essentially, you start with a
base number, called Zero, and then you have an operation, called
something like Successor, and the set of the Natural Numbers is the set
of things you get by applying that operation an unlimited number of times.

Thus, I admit to other number theories, they just are not the "standard"
systems of Natural Numbers, Integers, Rational Numbers, or Real Numbers.
(The Reals have a couple of different generational methods, but they
result in the same set of numbers with the same basic properties).

If you want to talk about other systems, feel free, just don't be
"deceptive" (or lying) and try to use the name for one of the standard sets.

And most of the "alternative" systems alternative have accepted names
for them, so that should be used.

[Ross Finlayson]

Post by Richard Damon

Post by Ross Finlayson

Post by Richard Damon

Post by WM

Post by Jim Burns
Calculus and WM's dark.number game
make a poor fit together.

Of course. Calculus uses potential infinity only. Dark numbers can exist
only in actual infinity, the complement of potential infinity.
Feferman's book In the light of logic answers the question: Is Cantor
necessary for the maths of the real world? with a resounding no.
Regards, WM

But you claim them to be part of the sets that are only "Potential
Infinity", or are you retracting that claim. (All Natural Numbers are
only by your definition "Potentially Infinite")
All members of the Natural Numbers are Finite, so none of them are
"Actually Infinite".
The set of them has a SIZE that is infinite, but none of the members of
it ever are.

The order type of ordinals is an ordinal.
Finite ordinals as the classes of each those not containing themselves,
according to simple quantification or "the set of all sets that don't
contain themselves", contains itself.
Such are simple reasons why the infinite integers have an infinite integer.
Yes I know that ZF two axioms of restriction of comprehension or "don't look"
include one that is "there's an infinity that isn't thusly various".
So, you might find it interesting that in more naive theories,
it's those one-line proofs as above that do give that there are
true infinites courtesy the unbounded its quantification.
Then about MW's "I invoke the Absolute!", it's he's drunk
then also that potential and actual infinities do have differences
in real mathematics, one recalls "A Sober Mind Speaks".
So, RD's "there is only one theory and it ignores all my problems"
and MWs "there are at least two theories and they don't agree",
seems for a sort of "yet somehow, there's a theory".
... With laws, plural, of large numbers.
If you want a gentle introduction to set theory,
try reading Quine's Set Theory then after it introduces
the class-set distinction and x not in x and x not equals x,
and why neither of those is first-order in ZF, quit.
... in a theory, ....

I understand the multitude of different ways to define "Numbers", but
their are a core set of fundamental numbers with basically agreed on
definitions, and reasonable people don't try calling something that
doesn't match the accepted definition as being one of those sets.
"Natural Numbers" is such a set, and while there may be several ways of
expressing its "generation", they all are essentially, you start with a
base number, called Zero, and then you have an operation, called
something like Successor, and the set of the Natural Numbers is the set
of things you get by applying that operation an unlimited number of times.
Thus, I admit to other number theories, they just are not the "standard"
systems of Natural Numbers, Integers, Rational Numbers, or Real Numbers.
(The Reals have a couple of different generational methods, but they
result in the same set of numbers with the same basic properties).
If you want to talk about other systems, feel free, just don't be
"deceptive" (or lying) and try to use the name for one of the standard sets.
And most of the "alternative" systems alternative have accepted names
for them, so that should be used.

Well, yes that's one of the maxims of language, that it's not abused.

That's one of the greatest claims against MW by old Virgil, again.
He doesn't know his words, which for example our professorial
James B. distinguishes notationally.

Then, in logic, and not just "an introduction to" but "the theory of"
and furthermore "foundations of", then there's addressed ambiguity,
inherent in definition, like quantifier disambiguation for Heap and Sorites,
like impredicativity for Heap and Sorites, like the sublime for Heap and Sorites,
the extra-ordinary for Heap and Sorites, and so on.

At the same time, or yet, ignorance either way is an abuse thereof, of language.
It's an insult to it.

Things like conflation for Heap and Sorites, invalid metaphor for Heap and Sorites,
false supposition for Heap and Sorites, these aren't proper conflation, metaphor,
and supposition, which are framed modally in contingents, simile, and fusion,
theories thereof, and therein.


So, I agree with what you wrote, but also sort of demand you acknowledge,
that the extra-ordinary, exists from lesser of its independent fundamental
principles, so rather precedes the otherwise usual formalism of "a regularity".

The regularity or well-foundedness understood as being made first order
in ZF set theory, and as noted above it puts membership the relation instead
of element-of the relation, out of first-order and so strictly intensional,
along though with equality or the fundamentally intensional.

Thus, class/set distinction, is a good example of when definitions go bad
from because axioms went, "wrong". The "principle axiom distinctions",
sort of mold theory following throughout.

That said, a dialectic in the Hegelian sense, thesis antithesis synthesis,
shows for "descriptive dynamics", that such as these vagaries or the vague,
are tussled out according to their definition and refinement and respect of language.

What's a thesis, ....

[Ross Finlayson]

Post by Ross Finlayson

Post by Richard Damon

Post by Ross Finlayson

Post by Richard Damon

Post by WM

Post by Jim Burns
Calculus and WM's dark.number game
make a poor fit together.

Of course. Calculus uses potential infinity only. Dark numbers can exist
only in actual infinity, the complement of potential infinity.
Feferman's book In the light of logic answers the question: Is Cantor
necessary for the maths of the real world? with a resounding no.
Regards, WM

But you claim them to be part of the sets that are only "Potential
Infinity", or are you retracting that claim. (All Natural Numbers are
only by your definition "Potentially Infinite")
All members of the Natural Numbers are Finite, so none of them are
"Actually Infinite".
The set of them has a SIZE that is infinite, but none of the members of
it ever are.

The order type of ordinals is an ordinal.
Finite ordinals as the classes of each those not containing themselves,
according to simple quantification or "the set of all sets that don't
contain themselves", contains itself.
Such are simple reasons why the infinite integers have an infinite integer.
Yes I know that ZF two axioms of restriction of comprehension or "don't look"
include one that is "there's an infinity that isn't thusly various".
So, you might find it interesting that in more naive theories,
it's those one-line proofs as above that do give that there are
true infinites courtesy the unbounded its quantification.
Then about MW's "I invoke the Absolute!", it's he's drunk
then also that potential and actual infinities do have differences
in real mathematics, one recalls "A Sober Mind Speaks".
So, RD's "there is only one theory and it ignores all my problems"
and MWs "there are at least two theories and they don't agree",
seems for a sort of "yet somehow, there's a theory".
... With laws, plural, of large numbers.
If you want a gentle introduction to set theory,
try reading Quine's Set Theory then after it introduces
the class-set distinction and x not in x and x not equals x,
and why neither of those is first-order in ZF, quit.
... in a theory, ....

I understand the multitude of different ways to define "Numbers", but
their are a core set of fundamental numbers with basically agreed on
definitions, and reasonable people don't try calling something that
doesn't match the accepted definition as being one of those sets.
"Natural Numbers" is such a set, and while there may be several ways of
expressing its "generation", they all are essentially, you start with a
base number, called Zero, and then you have an operation, called
something like Successor, and the set of the Natural Numbers is the set
of things you get by applying that operation an unlimited number of times.
Thus, I admit to other number theories, they just are not the "standard"
systems of Natural Numbers, Integers, Rational Numbers, or Real Numbers.
(The Reals have a couple of different generational methods, but they
result in the same set of numbers with the same basic properties).
If you want to talk about other systems, feel free, just don't be
"deceptive" (or lying) and try to use the name for one of the standard sets.
And most of the "alternative" systems alternative have accepted names
for them, so that should be used.

Well, yes that's one of the maxims of language, that it's not abused.
That's one of the greatest claims against MW by old Virgil, again.
He doesn't know his words, which for example our professorial
James B. distinguishes notationally.
Then, in logic, and not just "an introduction to" but "the theory of"
and furthermore "foundations of", then there's addressed ambiguity,
inherent in definition, like quantifier disambiguation for Heap and Sorites,
like impredicativity for Heap and Sorites, like the sublime for Heap and Sorites,
the extra-ordinary for Heap and Sorites, and so on.
At the same time, or yet, ignorance either way is an abuse thereof, of language.
It's an insult to it.
Things like conflation for Heap and Sorites, invalid metaphor for Heap and Sorites,
false supposition for Heap and Sorites, these aren't proper conflation, metaphor,
and supposition, which are framed modally in contingents, simile, and fusion,
theories thereof, and therein.
So, I agree with what you wrote, but also sort of demand you acknowledge,
that the extra-ordinary, exists from lesser of its independent fundamental
principles, so rather precedes the otherwise usual formalism of "a regularity".
The regularity or well-foundedness understood as being made first order
in ZF set theory, and as noted above it puts membership the relation instead
of element-of the relation, out of first-order and so strictly intensional,
along though with equality or the fundamentally intensional.
Thus, class/set distinction, is a good example of when definitions go bad
from because axioms went, "wrong". The "principle axiom distinctions",
sort of mold theory following throughout.
That said, a dialectic in the Hegelian sense, thesis antithesis synthesis,
shows for "descriptive dynamics", that such as these vagaries or the vague,
are tussled out according to their definition and refinement and respect of language.
What's a thesis, ....

Being that you demonstrate correct reasoning like a solid academic,
then in "foundations", which is the study overall of theory, and that
there is one, I hope you'll find it of interest that in the wider dialectic,
in the alatheia and dialatheia, the discovery of theory, that there really
is a lot to be garnered from the canon, that for most all history in the
Western canon is as about "Platonism", that there exist the ideals,
and we only discover them, then about all the usual things about the
reasoning, for things like "theories of relations and those to the binary
being predicates", like "a theory with one relation named elt called set theory".

So, "strong mathematical platonism" starts pretty much with "nothing, yet reason".

Here are a bunch podcasts, starting with "Philosophical Foreground", then
have a sort of handful of developments about descriptive differential dynamics,
then some readings from the applied like relativity, quantum mechanics,
and modern approaches to the non-linear.

https://www.youtube.com/@rossfinlayson

Then, I'm not quite sure yet how to just print all my posts to Usenet in order,
with or without the context, they form a sort of linear narrative,
described as "researches in foundations and requirements and desiderata thereof".

That there is one, ....


Yes, after Comte's Boole's Russell's logical positivism's Quine's, with respect
to Cantor's, or Zermelo and Fraenkel's, and Goedel's, a usual course for set theory,
here there's noted requirements of a sort of stronger theory, from less,
a principled while non-axiomatic approach, where stipulations result the contingent,
and logic is constant, complete, consistent, and concrete, that we attain to.

Warm regards, you describe some impeccable reasonings, but for example as
the Stanford Encyclopedia of Philosophy says for Heap and Sorites, it's important
that natural language is fulfilled, its correctness.

The theory is extra-ordinary, and whether it's ordering first or counting first,
and the other usual sort of notions of _regularity_ and the _rulial_, that
it's only one theory where all gets along.

Warm regards

[WM]

Post by Jim Burns

Post by WM
Just today I showed some proofs to my students.

If you are claiming your students for
some form of peer.review, then
you are accepting pre-calculus students as
_peers_

Nonsense. There are 8th semester informatics and engineering studens. They
know mathematics very well (that which Cantor is not needed for, according
to Feferman) but they do not use matheology. Try to show how the player
gets bankrupt in the game "We conquer the Binary Tree".

Regards, WM

[Jim Burns]

Post by WM

Post by Jim Burns

Post by WM
Just today I showed some proofs
to my students.

If you are claiming your students for
some form of peer.review, then
you are accepting pre-calculus students as
_peers_

Nonsense.
There are 8th semester informatics and
engineering studens.

Perhaps
the informatics and engineering students of
Wolfgang Mückenheim of Hochschule Augsburg
remember their arithmetic.

kᵢⱼ = i+(i+j-1)(i+j-2)/2

For each pair ⟨i,j⟩ of visibleᵂᴹ numbers
there is a unique visibleᵂᴹ number kᵢⱼ

sₖᵢⱼ = max{h| (h-1)(h-2)/2 < kᵢⱼ}
iₖᵢⱼ+jₖᵢⱼ = sₖᵢⱼ
iₖᵢⱼ = kᵢⱼ-(sₖᵢⱼ-1)(sₖᵢⱼ-2)/2
jₖᵢⱼ = sₖᵢⱼ(sₖᵢⱼ-1)/2-kᵢⱼ+1

For each visibleᵂᴹ number kᵢⱼ
there is a unique pair ⟨iₖᵢⱼ,jₖᵢⱼ⟩ of
visibleᵂᴹ numbers
⟨iₖᵢⱼ,jₖᵢⱼ⟩ = ⟨i,j⟩

There are as many visibleᵂᴹ numbers
as pairs of visibleᵂᴹ numbers.
Darkᵂᴹ numbers haven't entered the discussion.

----
For each non-empty split F,H of ⟨0,1,…,n⟩
some i‖i⁺¹ exists last‖first in F‖H
F‖H = ⟨0,1,…,i⟩‖⟨i⁺¹,…,n⟩

0‖n exists first‖last in ⟨0,1,…,n⟩

⟨0,1,…,n⟩ can't fit in ⟨1,…,n⟩
⟨0,1,…,n⟩ holds only visibleᵂᴹ numbers.

ℕ contains each ⟨0,1,…,n⟩ and nothing else.
ℕ\{0} contains each ⟨1,…,n⟩ and nothing else.

⟨0,1,…,n⟩ CAN fit in ⟨1,…,n,n⁺¹⟩
ℕ CAN fit in ℕ\{0}

If
darkᵂᴹ numbers are the reason that
ℕ CAN fit in ℕ\{0}
then
they must be
darkᵂᴹ visibleᵂᴹ in ⟨0,1,…,n⟩ in ℕ

In WM's scheme,
_sets_ inform _numbers_ that
the numbers are darkᵂᴹ
but we're not permitted to refer to
these darkᵂᴹ visibleᵂᴹ numbers,
only to the sets they're in.

Talking about
numbers which are darkᵂᴹ
is really nothing more than talking about
sets holding numbers which are darkᵂᴹ

The only use talking about darkᵂᴹ numbers has
is as a bit of techno.babble
to fill gaps in the discussion where,
otherwise, explanation would be expected.

[Ross Finlayson]

Post by Jim Burns

Post by WM

Post by Jim Burns

Post by WM
Just today I showed some proofs
to my students.

If you are claiming your students for
some form of peer.review, then
you are accepting pre-calculus students as
_peers_

Nonsense.
There are 8th semester informatics and
engineering studens.

Perhaps
the informatics and engineering students of
Wolfgang Mückenheim of Hochschule Augsburg
remember their arithmetic.
kᵢⱼ = i+(i+j-1)(i+j-2)/2
For each pair ⟨i,j⟩ of visibleᵂᴹ numbers
there is a unique visibleᵂᴹ number kᵢⱼ
sₖᵢⱼ = max{h| (h-1)(h-2)/2 < kᵢⱼ}
iₖᵢⱼ+jₖᵢⱼ = sₖᵢⱼ
iₖᵢⱼ = kᵢⱼ-(sₖᵢⱼ-1)(sₖᵢⱼ-2)/2
jₖᵢⱼ = sₖᵢⱼ(sₖᵢⱼ-1)/2-kᵢⱼ+1
For each visibleᵂᴹ number kᵢⱼ
there is a unique pair ⟨iₖᵢⱼ,jₖᵢⱼ⟩ of
visibleᵂᴹ numbers
⟨iₖᵢⱼ,jₖᵢⱼ⟩ = ⟨i,j⟩
There are as many visibleᵂᴹ numbers
as pairs of visibleᵂᴹ numbers.
Darkᵂᴹ numbers haven't entered the discussion.
----
For each non-empty split F,H of ⟨0,1,…,n⟩
some i‖i⁺¹ exists last‖first in F‖H
F‖H = ⟨0,1,…,i⟩‖⟨i⁺¹,…,n⟩
0‖n exists first‖last in ⟨0,1,…,n⟩
⟨0,1,…,n⟩ can't fit in ⟨1,…,n⟩
⟨0,1,…,n⟩ holds only visibleᵂᴹ numbers.
ℕ contains each ⟨0,1,…,n⟩ and nothing else.
ℕ\{0} contains each ⟨1,…,n⟩ and nothing else.
⟨0,1,…,n⟩ CAN fit in ⟨1,…,n,n⁺¹⟩
ℕ CAN fit in ℕ\{0}
If
darkᵂᴹ numbers are the reason that
ℕ CAN fit in ℕ\{0}
then
they must be
darkᵂᴹ visibleᵂᴹ in ⟨0,1,…,n⟩ in ℕ
In WM's scheme,
_sets_ inform _numbers_ that
the numbers are darkᵂᴹ
but we're not permitted to refer to
these darkᵂᴹ visibleᵂᴹ numbers,
only to the sets they're in.
Talking about
numbers which are darkᵂᴹ
is really nothing more than talking about
sets holding numbers which are darkᵂᴹ
The only use talking about darkᵂᴹ numbers has
is as a bit of techno.babble
to fill gaps in the discussion where,
otherwise, explanation would be expected.

Another word for explanations is "apologetics",
then here it's disambiguation, because in mathematics
there's an overall sort of teleological approach following
after any ontological approach, because matheamtics
is constant in relation. Then, "strong Platonism" has that
it's overall teleological, and as down from Kant's sublime and
why Man and Mind have an object-sense, number-sense, word-sense,
to complement the usual phenomenal senses, and as what's thought.

Your sort of exercise in delineating MW's definitions, is a great
sort of approach, because it keeps then those from occluding
the, "real", definitions.

I.e. it's a reasonable sort of didactic approach,
to remove confusion from conflation.

Then, the idea that there are "descriptive dynamics",
has that basically all theories formally abstractly symbolically
result sorts of descriptive theories in words, all one theory.

I enjoy your writings and hope for success in your endeavors.

[Jim Burns]

On Thursday, January 11, 2024

[...]

Then, the idea that there are
"descriptive dynamics",
has that
basically all theories
formally abstractly symbolically
result sorts of descriptive theories
in words, all one theory.

There is a Swiss.Army.knife theory, which
provides insight in many, many instances.

I'm thinking of the theory of
finite sequences of claims,
claims which might be about any of
many, many different domains of discussion.

We have a lot of confidence that,
in a finite sequence of claims,
if there is a false claim,
then there is a first.false claim.

Join that to our ability to _see_
in some sequences,
that some claims are not.first.false,
such as Q in ⟨ Q∨¬P P Q ⟩
and
we know we have, in some instances,
finite sequences of not.first.false claims,
which,
by our Swiss.Army.knife theory,
are finite sequences of not.false claims.

Even if
we can't see the objects of our claims,
we know they are true claims.
That seems magical to me --
"magical" perhaps not the best word for
mathematics, science, and engineering,
but there it is.

This Swiss.Army.knife theory of
finite sequences of claims
might be something like
what you (RF) mean by One Theory.
Or it might not be.

----
An essential step in using the Swiss.Army.knife
is description.
We decide to learn about widgets.
_We describe a widget_
We follow with not.first.false claims
about widgets, which,
because they are in that sequence,
we know are true about widgets.
Success.

That works because there are
pre.known claims about widgets mixed in.
That makes our post.known claims
specific to widgets _which we want_
since widgets are different from non.widgets.

But the pre.known claims are widget theory,
not the One Theory.
If we replace claims about widgets with
claims about anything,
following claims not.first.false about anything
shouldn't be telling us anything useful or
interesting, should it?
I don't see how it could.

[WM]

Post by Jim Burns

Post by WM

Post by Jim Burns

Post by WM
Just today I showed some proofs
to my students.

If you are claiming your students for
some form of peer.review, then
you are accepting pre-calculus students as
_peers_

Nonsense.
There are 8th semester informatics and
engineering studens.

Perhaps
the informatics and engineering students of
Wolfgang Mückenheim of Hochschule Augsburg
remember their arithmetic.
kᵢⱼ = i+(i+j-1)(i+j-2)/2

Every number defined like k does not belong to the domain covered by the
smallest
ℵo unit fractions or by the largest ℵo natural numbers.

Regards, WM

[Jim Burns]

Post by WM

Post by Jim Burns

Post by WM

Post by Jim Burns

Post by WM
Just today I showed some proofs
to my students.

If you are claiming your students for
some form of peer.review, then
you are accepting pre-calculus students as
_peers_

Nonsense.
There are 8th semester informatics and
engineering studens.

Perhaps
the informatics and engineering students of
Wolfgang Mückenheim of Hochschule Augsburg
remember their arithmetic.
kᵢⱼ = i+(i+j-1)(i+j-2)/2

Every number defined like k
does not belong to the domain covered by
the smallest ℵo unit fractions or by
the largest ℵo natural numbers.

Perhaps
the informatics and engineering students of
Wolfgang Mückenheim of Hochschule Augsburg
remember that,
in arithmetic,
the unit fractions and natural numbers
are closed under addition, subtraction,
multiplication, and division.

Since you (WM) have decided that
you are talking about _not.arithmetic_
this might be an especially apt time
for your students to remember that
a claim about _not.arithmetic_
even if it were _true_
doesn't contradict a claim about _arithmetic_

[FromTheRafters]

Post by Jim Burns

Post by WM

Post by Jim Burns

Post by WM

Post by Jim Burns

Post by WM
Just today I showed some proofs
to my students.

If you are claiming your students for
some form of peer.review, then
you are accepting pre-calculus students as
_peers_

Nonsense.
There are 8th semester informatics and
engineering studens.

Perhaps
the informatics and engineering students of
Wolfgang Mückenheim of Hochschule Augsburg
remember their arithmetic.
kᵢⱼ = i+(i+j-1)(i+j-2)/2

Every number defined like k
does not belong to the domain covered by
the smallest ℵo unit fractions or by
the largest ℵo natural numbers.

Perhaps
the informatics and engineering students of
Wolfgang Mückenheim of Hochschule Augsburg
remember that,
in arithmetic,
the unit fractions and natural numbers
are closed under addition, subtraction,
multiplication, and division.
Since you (WM) have decided that
you are talking about _not.arithmetic_
this might be an especially apt time
for your students to remember that
a claim about _not.arithmetic_
even if it were _true_
doesn't contradict a claim about _arithmetic_

Subtraction?

[Jim Burns]

Post by FromTheRafters

Post by Jim Burns

Post by WM

Post by Jim Burns
kᵢⱼ = i+(i+j-1)(i+j-2)/2

Every number defined like k
does not belong to the domain covered by
the smallest ℵo unit fractions or by
the largest ℵo natural numbers.

Perhaps
the informatics and engineering students of
Wolfgang Mückenheim of Hochschule Augsburg
remember that,
in arithmetic,
the unit fractions and natural numbers
are closed under addition, subtraction,
multiplication, and division.
Since you (WM) have decided that
you are talking about _not.arithmetic_
this might be an especially apt time
for your students to remember that
a claim about _not.arithmetic_
even if it were _true_
doesn't contradict a claim about _arithmetic_

Subtraction?

I may have been a little loose in
how I said that.

The operations are closed in _arithmetic_
except for division by 0.

There is
no single ℵ₀.set of smallest unit fractions and
no single ℵ₀.set of largest natural numbers.

There is,
for each pair ⟨i,j⟩ of visibleᵂᴹ numbers
a unique visibleᵂᴹ number kᵢⱼ
and,
for each visibleᵂᴹ number kᵢⱼ
a unique pair ⟨iₖᵢⱼ,jₖᵢⱼ⟩ of visibleᵂᴹ numbers
⟨iₖᵢⱼ,jₖᵢⱼ⟩ = ⟨i,j⟩

There are as many visibleᵂᴹ numbers
as pairs of visibleᵂᴹ numbers.
Darkᵂᴹ numbers haven't entered the discussion.

kᵢⱼ = i+(i+j-1)(i+j-2)/2

sₖᵢⱼ = max{h| (h-1)(h-2)/2 < kᵢⱼ}
iₖᵢⱼ+jₖᵢⱼ = sₖᵢⱼ
iₖᵢⱼ = kᵢⱼ-(sₖᵢⱼ-1)(sₖᵢⱼ-2)/2
jₖᵢⱼ = sₖᵢⱼ(sₖᵢⱼ-1)/2-kᵢⱼ+1

Because arithmetic.

[Fritz Feldhase]

for each pair ⟨i,j⟩ of [natural] numbers
[there is] a unique [natural] number kᵢⱼ
and,
for each [natural] number kᵢⱼ [there is]
a unique pair ⟨iₖᵢⱼ,jₖᵢⱼ⟩ of [natural] numbers
⟨iₖᵢⱼ,jₖᵢⱼ⟩ = ⟨i,j⟩

Sure.

There are as many [natural] numbers as pairs of [natural] numbers.

What we actually know is that there is bijection between the (set of) natural numbers and the (set of) pairs of natural numbers.

For deriving "there are as many X as Y" we need more than just that.

Note that WM claims:

1. There are no bijections between (different) infinite sets. (wrong in the context of set theory)

2. Even if there were a bijection between two infinites sets A and B, this does NOT mean, that A and B are equinumerous. (wrong in the context of set theory)

*sigh*

P. Suppes (temporarily) adopted an axiom for his system of set theory which stated: card(A) = card(B) iff A ~ B. This statement (either as an axiom or a theorem) is pretty much what we need to conclude card(A) = card(A) from A ~ B, allowing us to call A and B equinumerous.

[Ross Finlayson]

Post by Jim Burns

Post by FromTheRafters

Post by Jim Burns

Post by WM

Post by Jim Burns
kᵢⱼ = i+(i+j-1)(i+j-2)/2

Every number defined like k
does not belong to the domain covered by
the smallest ℵo unit fractions or by
the largest ℵo natural numbers.

Perhaps
the informatics and engineering students of
Wolfgang Mückenheim of Hochschule Augsburg
remember that,
in arithmetic,
the unit fractions and natural numbers
are closed under addition, subtraction,
multiplication, and division.
Since you (WM) have decided that
you are talking about _not.arithmetic_
this might be an especially apt time
for your students to remember that
a claim about _not.arithmetic_
even if it were _true_
doesn't contradict a claim about _arithmetic_

Subtraction?

I may have been a little loose in
how I said that.
The operations are closed in _arithmetic_
except for division by 0.
There is
no single ℵ₀.set of smallest unit fractions and
no single ℵ₀.set of largest natural numbers.
There is,
for each pair ⟨i,j⟩ of visibleᵂᴹ numbers
a unique visibleᵂᴹ number kᵢⱼ
and,
for each visibleᵂᴹ number kᵢⱼ
a unique pair ⟨iₖᵢⱼ,jₖᵢⱼ⟩ of visibleᵂᴹ numbers
⟨iₖᵢⱼ,jₖᵢⱼ⟩ = ⟨i,j⟩
There are as many visibleᵂᴹ numbers
as pairs of visibleᵂᴹ numbers.
Darkᵂᴹ numbers haven't entered the discussion.
kᵢⱼ = i+(i+j-1)(i+j-2)/2
sₖᵢⱼ = max{h| (h-1)(h-2)/2 < kᵢⱼ}
iₖᵢⱼ+jₖᵢⱼ = sₖᵢⱼ
iₖᵢⱼ = kᵢⱼ-(sₖᵢⱼ-1)(sₖᵢⱼ-2)/2
jₖᵢⱼ = sₖᵢⱼ(sₖᵢⱼ-1)/2-kᵢⱼ+1
Because arithmetic.

What you can do is define arithmetic just a little differently than usual, that instead
of Peano/Presburger, addition and multiplication, and their inverses, is that what
you do is start with increment and division.

Then there's as
increment addition
multiplication powers
exponent tetration
and these kinds paired powers under addition formulae that result, and
division
subtraction
decrement
as it sort of works out, that instead of, "long division", there's, "long subtraction".


Then, for models of arithmetic that are standard, they're about the same, but
it's when the models are nonstandard as bounded fragments, that
"increment and division" keep it regular and rulial both ways,
that it's regular and rulial both ways, instead of ragged.

I.e., in the models of algebras of arithmetics and arithmetics of algebras, that
not only are algebra and arithmetic separated, it's shown how they come together.


Then, division by zero is a singularity, in singularity theory, in what's called multiplicity theory.
In such theories, division-by-zero may or may not be "indeterminate forms" vis-a-vis either
"Not a Number" or "undefined.

In my podcasts in "descriptive differential dynamics", I introduce concepts after some
of these concepts of arithmetic and algebra, after "roots of zero", into roots of unity,
then about the "identity dimension", about where x=y is a singularity in many common
systems of differential equations, and included or not just like zero is, and a bunch of
different interesting features of arithmetic and algebra and their common field.

It's numbers, and letters, ....

[Jim Burns]

On Thursday, January 11, 2024

Post by Jim Burns
There are as many visibleᵂᴹ numbers
as pairs of visibleᵂᴹ numbers.
Darkᵂᴹ numbers haven't entered the discussion.
kᵢⱼ = i+(i+j-1)(i+j-2)/2
sₖᵢⱼ = max{h| (h-1)(h-2)/2 < kᵢⱼ}
iₖᵢⱼ+jₖᵢⱼ = sₖᵢⱼ
iₖᵢⱼ = kᵢⱼ-(sₖᵢⱼ-1)(sₖᵢⱼ-2)/2
jₖᵢⱼ = sₖᵢⱼ(sₖᵢⱼ-1)/2-kᵢⱼ+1
Because arithmetic.

What you can do is
define arithmetic just a little
differently than usual, that instead
of Peano/Presburger,
addition and multiplication,
and their inverses,
is that what you do is
start with increment and division.

Julia Robinson has a paper showing how
to define various arithmetical concepts
from each other.

Definability and decision problems in arithmetic

| In this paper, we are concerned with
| the arithmetical definability of certain notions
| of integers and rationals in terms of other notions.
| The results derived will be applied to obtain
| a negative solution of corresponding decision problems.
|
| In Section 1, we show that addition of
| positive integers can be defined arithmetically
| in terms of multiplication and the unary operation
| of successor S (where Sa = a + 1). Also, it is shown
| that both addition and multiplication can be defined
| arithmetically in terms of successor and
| the relation of divisibility |
| (where x|y means x divides y).
|
The Journal of Symbolic Logic,
Volume 14, Issue 2, 23 June 1949, pp. 98 - 114
DOI: https://doi.org/10.2307/2266510

That is also interesting because
some claims become easier or harder to prove,
starting from order and divisibility or
from addition and multiplication.

If I recall correctly,
unique prime factorization becomes easier
starting from order and divisibility.

Then there's as
increment addition
multiplication powers
exponent tetration

Once we have addition and multiplication,
however we get them,
the Chinese remainder theorem can be used
to encode an arbitrarily.long finite sequence,
and encoding finite sequences can be used
to re.phrase recursive definitions
non.recursively.

That makes exponentials, factorials,
tetration, and much, much more
Free At No Extra Charge.

Just saying.

[Fritz Feldhase]

Post by FromTheRafters

Post by Jim Burns

Post by WM

Post by Jim Burns

Post by WM

Post by Jim Burns

Post by WM
Just today I showed some proofs
to my students.

If you are claiming your students for
some form of peer.review, then
you are accepting pre-calculus students as
_peers_

Nonsense.
There are 8th semester informatics and
engineering studens.

Perhaps
the informatics and engineering students of
Wolfgang Mückenheim of Hochschule Augsburg
remember their arithmetic.
kᵢⱼ = i+(i+j-1)(i+j-2)/2

Every number defined like k
does not belong to the domain covered by
the smallest ℵo unit fractions or by
the largest ℵo natural numbers.

Perhaps
the informatics and engineering students of
Wolfgang Mückenheim of Hochschule Augsburg
remember that,
in arithmetic,
the unit fractions and natural numbers
are closed under addition, subtraction,
multiplication, and division.
Since you (WM) have decided that
you are talking about _not.arithmetic_
this might be an especially apt time
for your students to remember that
a claim about _not.arithmetic_
even if it were _true_
doesn't contradict a claim about _arithmetic_

Subtraction?

Division? :-P

[WM]

Post by Jim Burns
Since you (WM) have decided that
you are talking about _not.arithmetic_
this might be an especially apt time
for your students to remember that
a claim about _not.arithmetic_
even if it were _true_
doesn't contradict a claim about _arithmetic_

That should not hinder an inquisitive student to learn that arithmetic
does not cover the domain of the smallest ℵo unit fractions and of the
largest ℵo natural numbers.

Regards, WM

[Jim Burns]

Post by WM

Post by Jim Burns
Since you (WM) have decided that
you are talking about _not.arithmetic_
this might be an especially apt time
for your students to remember that
a claim about _not.arithmetic_
even if it were _true_
doesn't contradict a claim about _arithmetic_

That should not hinder an inquisitive student
to learn that
arithmetic does not cover the domain of
the smallest ℵo unit fractions and of
the largest ℵo natural numbers.

Do your students learn that it doesn't from
their instructor, from you?

If they don't,
that is what strikes me as
hindering the inquisitive student.

----
We can make a claim true of
each one of infinitely.many
which we know is true by knowing
_that is what we mean_ by
"natural number" or "unit fraction"
or something else.

Also,
we can make a claim not.first.false of
each one of the same infinitely.many
which we know is not.first.false by seeing
the sequence of claims we make.
For example, by seeing Q is after P and Q∨¬P

We can make a finite sequence of claims
in which each claim is not.first.false of
each one of the same infinitely.many.
This is more challenging, but the challenge
is the reason that mathematicians are
so famously well-paid.

A finite sequence in which
each claim is not.first.false about each
is finite sequence in which
each claim is not.false about each.

That is something we already know
_about finite sequences of claims_

By that knowledge,
we can increase our knowledge of other things:
natural numbers, unit fractions, and so on
_looking only at a sequence of claims_
and NOT looking at
natural numbers, unit fractions, and so on.

----
We can make a claim that
k can be counted to
and know that it is true of
each which can be counted to.

We can follow that with the claim that
k+1 can be counted to
and see that there is no way in which
k+1 can be counted to
can be first.false.

And, similarly, that
k isn't the largest which can be counted to.

That is how we know that
nothing exists which is the largest which
can be counted to,
even though we are finite beings,
even though we can't see the infinitely.many
which can be counted to.

We can see _the claims_
and recognize that they're not.first.false.

[Ross Finlayson]

Post by Jim Burns

Post by WM

Post by Jim Burns
Since you (WM) have decided that
you are talking about _not.arithmetic_
this might be an especially apt time
for your students to remember that
a claim about _not.arithmetic_
even if it were _true_
doesn't contradict a claim about _arithmetic_

That should not hinder an inquisitive student
to learn that
arithmetic does not cover the domain of
the smallest ℵo unit fractions and of
the largest ℵo natural numbers.

Do your students learn that it doesn't from
their instructor, from you?
If they don't,
that is what strikes me as
hindering the inquisitive student.
----
We can make a claim true of
each one of infinitely.many
which we know is true by knowing
_that is what we mean_ by
"natural number" or "unit fraction"
or something else.
Also,
we can make a claim not.first.false of
each one of the same infinitely.many
which we know is not.first.false by seeing
the sequence of claims we make.
For example, by seeing Q is after P and Q∨¬P
We can make a finite sequence of claims
in which each claim is not.first.false of
each one of the same infinitely.many.
This is more challenging, but the challenge
is the reason that mathematicians are
so famously well-paid.
A finite sequence in which
each claim is not.first.false about each
is finite sequence in which
each claim is not.false about each.
That is something we already know
_about finite sequences of claims_
By that knowledge,
natural numbers, unit fractions, and so on
_looking only at a sequence of claims_
and NOT looking at
natural numbers, unit fractions, and so on.
----
We can make a claim that
k can be counted to
and know that it is true of
each which can be counted to.
We can follow that with the claim that
k+1 can be counted to
and see that there is no way in which
k+1 can be counted to
can be first.false.
And, similarly, that
k isn't the largest which can be counted to.
That is how we know that
nothing exists which is the largest which
can be counted to,
even though we are finite beings,
even though we can't see the infinitely.many
which can be counted to.
We can see _the claims_
and recognize that they're not.first.false.

I've been following this guy for years.

His dialectic on the inductive impasse and deductive resolution is very good.

[Jim Burns]

It is obviously impossible
to come closer to zero.

No,
it is obvious in arithmetic
that each x ≠ 0 can come closer.
To x/2  for example.

There is no boundary in
the measure of distance
but
there is a boundary in
the measure of remaining elements
between 0 and the chosen eps > 0.
∀eps > 0, ∀x ∈ (eps, 1]: NUF(x) = ℵo

In arithmetic,
∀eps > 0
∀k ∈ ℕ
∃mᵉᵖˢ ∈ ℕ:
eps > ⅟mᵉᵖˢ > ⅟(mᵉᵖˢ+k+1) > 0
k < NUF(eps)

In arithmetic,
∀eps > 0
∀k ∈ ℕ
k < NUF(eps)
NUF(eps) ≮ NUF(eps)
NUF(eps) ∉ ℕ

In arithmetic,
∀eps > 0
NUF(eps) ∉ ℕ

In arithmetic,
¬∃mᵂᴹ ∈ ℕ:
∀eps > 0
∀k ∈ ℕ
eps > ⅟mᵂᴹ > ⅟(mᵂᴹ+k+1) > 0

there is a boundary in
the measure of remaining elements
between 0 and the chosen eps > 0.

In arithmetic,
there is a boundary in
the cardinality of remaining elements
between 0 and the chosen eps > 0.
but
it's not a positive boundary,
the boundary is 0

| Assume otherwise.
| Assume 0 < β/2 < β < 2β
| for remaining.element.card.boundary β
|
| β < 2β
| NUF(2β) ∉ ℕ
| ∀k ∈ ℕ
| ∃mₖ ∈ ℕ:
| 2β > ⅟mₖ > ⅟(mₖ+k+1) > 0
| [1]
| k < NUF(2β)
|
| β/2 < β
| NUF(β/2) ∈ ℕ
| ¬∀k ∈ ℕ
| ∃mₖ ∈ ℕ:
| β/2 > ⅟mₖ > ⅟(mₖ+k+1) > 0
|
| ∃k ∈ ℕ:
| ¬(k < NUF(β/2))
|
| ∃k ∈ ℕ:
| ∀m′ ∈ ℕ
| ¬(β/2 > ⅟m′ > ⅟(m′+k+1) > 0)
| β/2 ≤ ⅟m′ ∨ ¬(⅟m′ > ⅟(m′+k+1) > 0)
| ⅟m′ > ⅟(m′+k+1)) > 0
| β/2 ≤ ⅟m′
|
| ∀m′ ∈ ℕ
| β/2 ≤ ⅟m′
| [2]
|
| From [1]
| ∀k ∈ ℕ
| ∃mₖ ∈ ℕ:
| 2β > ⅟mₖ
| β/2 > ⅟(4mₖ)
|
| However, from [2]
| in the case of m′ = 4mₖ
| β/2 ≤ ⅟(4mₖ)
| Contradiction.

Therefore,
in arithmetic,
the remaining.element.measure.boundary β
isn't positive.

In arithmetic,
∀eps > 0:
NUF(eps) ∉ ℕ

[WM]

Post by Jim Burns
In arithmetic,
NUF(eps) ∉ ℕ

In arithmetic

∀x ∈ (0, 1]: y < x ==> y =< 0, i.e., y is not positive.

Regards, WM

[Richard Damon]

Post by Jim Burns

Post by Jim Burns
In arithmetic,
NUF(eps) ∉ ℕ

In arithmetic
∀x ∈ (0, 1]: y < x ==> y =< 0, i.e., y is not positive.
Regards, WM

Except that the statement does't hold if, say, y was x/2.

You also need to define the domain of x and y. If not specified it will
be presumed something like The Reals.

Your problem is you have the order of operations wrong in your logic,
the Qualifier is run first, so when we look at the inequality, we have
am x we can use.

[Dieter Heidorn]

Post by Richard Damon

Post by Jim Burns

Post by Jim Burns
In arithmetic,
NUF(eps) ∉ ℕ

In arithmetic
∀x ∈ (0, 1]: y < x ==> y =< 0, i.e., y is not positive.
Regards, WM

Except that the statement does't hold if, say, y was x/2.

That's right. But what WM means is the following:

for every y which is smaller than every x ∈ (0,1] : y <= 0.

One of his problems is that he doesn't understand quantifiers and their
proper application. Example: The correct statement about unit fractions
1/n (where n∈ℕ)

∀ x ∈ (0,1] ∃^ℵo 1/n : 1/n < x

he misunderstands in a way that is equivalent to the quantifier shift:

∃^ℵo 1/n ∀ x ∈ (0,1] : 1/n < x .

Then he concludes:

there are ℵo unit fractions left from zero.

Discussing with WM you should always keep in mind: WM doesn't write
about mathematics but his private system of ideas which are based on
his inability to understand infinite sets and set theory.

Dieter Heidorn

[Fritz Feldhase]

∀ x ∈ (0,1] ∃^ℵo 1/n : 1/n < x

∀ x ∈ (0,1]: ∃^ℵo n ∈ ℕ: 1/n < x ,

or even better:

∀ x ∈ (0,1]: ∃^ℵo u ∈ {1/n : n ∈ ℕ}: u < x

∃^ℵo 1/n ∀ x ∈ (0,1] : 1/n < x .

∃^ℵo n ∈ ℕ: ∀ x ∈ (0,1]: 1/n < x ,

or even better:

∃^ℵo u ∈ {1/n : n ∈ ℕ}: ∀ x ∈ (0,1]: u < x

Hint: Technically, in the context of a quantifier you can't use a "complex term" like "1/n" as if it were a /variable/. This is mandatory: ∀<variable> and ∃<variable>.

[WM]

Post by Dieter Heidorn
The correct statement about unit fractions
1/n (where n∈ℕ)
∀ x ∈ (0,1] ∃^ℵo 1/n : 1/n < x
∃^ℵo 1/n ∀ x ∈ (0,1] : 1/n < x .
there are ℵo unit fractions left from zero.

Would be required to make your statement correct.

For the less-than relation there is no quantifier magic.
NUF(x) cannot grow anywhere from 0 to ℵo without passing finite values.

Regards, WM

[Richard Damon]

Post by WM

Post by Dieter Heidorn
The correct statement about unit fractions
1/n (where n∈ℕ)
    ∀ x ∈ (0,1]  ∃^ℵo 1/n : 1/n < x
    ∃^ℵo 1/n  ∀ x ∈ (0,1] : 1/n < x .
    there are ℵo unit fractions left from zero.

Would be required to make your statement correct.
For the less-than relation there is no quantifier magic.
NUF(x) cannot grow anywhere from 0 to ℵo without passing finite values.
Regards, WM

Why not?

And what says those "values" that it happens at have to be in the set of
Unit Fractions/rational/real numbers?

If the "value" where NUF(x) == 1 isn't in the domain of Unit
factions/Rational Numbers/ Real Numbers, then there is no need for your
"dark" numbers, they are just visible numbers in some other system.

[WM]

Post by Richard Damon

Post by WM
NUF(x) cannot grow anywhere from 0 to ℵo without passing finite values.

Why not?

Because of mathemtics. ∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0.

Regards, WM

[Richard Damon]

Post by WM

Post by Richard Damon

Post by WM
NUF(x) cannot grow anywhere from 0 to ℵo without passing finite values.

Why not?

Because of mathemtics. ∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0.
Regards, WM

And how does that say that NUF(x) can't grow to ℵo without passing
finite values.

That equation in fact proves that there can not be a smallest 1/n as the
'level gap' below 1/n is only 1/(n+1) of the distance between 0 and 1/n,
so there is room for at least n+1 more unit fractions below it.

You need to find a point where the gap is as big as 1/n, and it never is.

[WM]

Post by Richard Damon

Post by WM

Post by Richard Damon

Post by WM
NUF(x) cannot grow anywhere from 0 to ℵo without passing finite values.

Why not?

Because of mathemtics. ∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0.

And how does that say that NUF(x) can't grow to ℵo without passing
finite values.

Because after every unit fraction the function NUF(x) is constant over d_n

Post by Richard Damon
0.
That equation in fact proves that there can not be a smallest 1/n as the
'level gap' below 1/n is only 1/(n+1) of the distance between 0 and 1/n,
so there is room for at least n+1 more unit fractions below it.

Nevertheless ***all*** unit fractions have gaps between each other. There
is no exception.

Post by Richard Damon
You need to find a point where the gap is as big as 1/n, and it never is.

We cannot investigate individuals within the dark domain.

Regards, WM

[FromTheRafters]

Post by WM

Post by WM

Post by Richard Damon

Post by WM
NUF(x) cannot grow anywhere from 0 to ℵo without passing finite values.

Why not?

Because of mathemtics. ∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0.

And how does that say that NUF(x) can't grow to ℵo without passing finite
values.

Because after every unit fraction the function NUF(x) is constant over d_n

0.
That equation in fact proves that there can not be a smallest 1/n as the
'level gap' below 1/n is only 1/(n+1) of the distance between 0 and 1/n, so
there is room for at least n+1 more unit fractions below it.

Nevertheless ***all*** unit fractions have gaps between each other. There is
no exception.

Yes, there are gaps in Q+ with respect to the positive reals fractional
parts. Unless "***all*** unit fractions" means the set of unit
fractions, in which case you just babble as usual.

[WM]

Post by FromTheRafters
Yes, there are gaps in Q+ with respect to the positive reals fractional
parts.

That means NUF(x) does not increase by more than 1 without stopping
afterwards. It starts with 0 and not with ℵ.

Regards, WM

[FromTheRafters]

Post by WM

Post by FromTheRafters
Yes, there are gaps in Q+ with respect to the positive reals fractional
parts.

That means NUF(x) does not increase by more than 1 without stopping
afterwards.

No, it doesn't.

Post by WM
It starts with 0 and not with ℵ.

It is not an action.

[WM]

Post by FromTheRafters

Post by WM

Post by FromTheRafters
Yes, there are gaps in Q+ with respect to the positive reals fractional
parts.

That means NUF(x) does not increase by more than 1 without stopping
afterwards.

No, it doesn't.

In mathematics it does. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0

Post by FromTheRafters

Post by WM
It starts with 0 and not with ℵ.

It is not an action.

Hogwash. It is a function.

Regards, WM

[FromTheRafters]

Post by WM

Post by FromTheRafters

Post by WM

Post by FromTheRafters
Yes, there are gaps in Q+ with respect to the positive reals fractional
parts.

That means NUF(x) does not increase by more than 1 without stopping
afterwards.

No, it doesn't.

In mathematics it does. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0

Post by FromTheRafters

Post by WM
It starts with 0 and not with ℵ.

It is not an action.

Hogwash. It is a function.

Then map it, done. No start here and finish there involved.

[Richard Damon]

Post by WM

Post by Richard Damon

Post by WM

Post by Richard Damon

Post by WM
NUF(x) cannot grow anywhere from 0 to ℵo without passing finite values.

Why not?

Because of mathemtics. ∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0.

And how does that say that NUF(x) can't grow to ℵo without passing
finite values.

Because after every unit fraction the function NUF(x) is constant over d_n

So? That doesn't mean that NUF(x) can't instantly grow to infinity
between 0 and the range (0,1]

Post by WM

Post by Richard Damon
0.
That equation in fact proves that there can not be a smallest 1/n as
the 'level gap' below 1/n is only 1/(n+1) of the distance between 0
and 1/n, so there is room for at least n+1 more unit fractions below it.

Nevertheless ***all*** unit fractions have gaps between each other.
There is no exception.

And again, we aren't talking BETWEEN unit fractions, but between 0 and
(0,1].

Post by WM

Post by Richard Damon
You need to find a point where the gap is as big as 1/n, and it never is.

We cannot investigate individuals within the dark domain.

You can not investigate the dark domain because it doesn't exist.

Post by WM
Regards, WM

[WM]

Post by Richard Damon

Post by WM
Because after every unit fraction the function NUF(x) is constant over d_n

So? That doesn't mean that NUF(x) can't instantly grow to infinity
between 0 and the range (0,1]

It does mean exactly this. Otherwise:

If for every point x > 0 there are ℵ smaller unit fractions,
then there exists no point x > 0 without ℵ smaller unit fractions.
Then there exist ℵ negative unit fractions.

Regards, WM

[Chris M. Thomasson]

Post by WM

Post by Richard Damon

Post by WM

Post by Richard Damon

Post by WM
NUF(x) cannot grow anywhere from 0 to ℵo without passing finite values.

Why not?

Because of mathemtics. ∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0.

And how does that say that NUF(x) can't grow to ℵo without passing
finite values.

Because after every unit fraction the function NUF(x) is constant over d_n

Post by Richard Damon
0.
That equation in fact proves that there can not be a smallest 1/n as
the 'level gap' below 1/n is only 1/(n+1) of the distance between 0
and 1/n, so there is room for at least n+1 more unit fractions below it.

Nevertheless ***all*** unit fractions have gaps between each other.
There is no exception.

Post by Richard Damon
You need to find a point where the gap is as big as 1/n, and it never is.

We cannot investigate individuals within the dark domain.

WM says this number is dark, well, WM just named it, so how dark is it?
Grey? lol... ;^)

[Jim Burns]

Post by WM

[...]

For the less-than relation
there is no quantifier magic.

For anti.symmetric relations, such as less.than,
there is quantifier anti.magic.

| Assume P(x,y) ⇔ ¬P(y,x)
| Assume ∀x:∃y:P(x,y)
|
| ∀x:∃y:P(x,y)
| if and only if
| ¬∃x:∀y:¬P(x,y)
| if and only if
| ¬∃x:∀y:P(y,x)
| relabel 'x''z','y''x','z''y'
| ¬∃y:∀x:P(x,y)

Therefore,
P(x,y) ⇔ ¬P(y,x)
∀x:∃y:P(x,y)
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
¬∃y:∀x:P(x,y)
anti.magic

∀x:∃y≠x: x<y
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
¬∃y:∀y≠x: x<y
anti.magic

[Fritz Feldhase]

Post by WM
For the less-than relation there is no quantifier magic.

Was immer Deine "quantifier magic" auch sein soll; aber die (richtige) Reihenfolge der Quantoren ist wesentlich:

An e IN: Em e IN: n < m (true)

Em e IN: An e IN: n < m (false)

[WM]

Post by Richard Damon

Post by Jim Burns

Post by Jim Burns
In arithmetic,
NUF(eps) ∉ ℕ

In arithmetic
∀x ∈ (0, 1]: y < x ==> y =< 0, i.e., y is not positive.

Except that the statement does't hold if, say, y was x/2.

The statement concerns all real numbers.

Post by Richard Damon
You also need to define the domain of x and y. If not specified it will
be presumed something like The Reals.
Your problem is you have the order of operations wrong in your logic,
the Qualifier is run first, so when we look at the inequality, we have
am x we can use.

The "less than" relation is independent of quantifier-exchange. If for
every x ∈ (0, 1] there a smaller y, then there is an y smaller than
every x ∈ (0, 1] and hence smaller than (0, 1]. Of course for ℵ
elements y this is much clearer.

Regards, WM

[Richard Damon]

Post by WM

Post by Richard Damon

Post by Jim Burns

Post by Jim Burns
In arithmetic,
NUF(eps) ∉ ℕ

In arithmetic
∀x ∈ (0, 1]: y < x ==> y =< 0, i.e., y is not positive.

Except that the statement does't hold if, say, y was x/2.

The statement concerns all real numbers.

Post by Richard Damon
You also need to define the domain of x and y. If not specified it
will be presumed something like The Reals.
Your problem is you have the order of operations wrong in your logic,
the Qualifier is run first, so when we look at the inequality, we have
am x we can use.

The "less than" relation is independent of quantifier-exchange. If for
every x ∈ (0, 1] there a smaller  y, then there is an y smaller than
every x ∈ (0, 1] and hence smaller than (0, 1]. Of course for ℵ elements
y this is much clearer.
Regards, WM

Right, for every x ∈ (0, 1] there exist a y such that 0 < y < x.

That doesn't mean that y < (0, 1], because that is a category error. x
and y were elements of the set and don't have a numeric relations to the
set (sets aren't numbers, but sets of numbers)

This is just expressing the unbounded nature of this set of numbers,
there is NO "lowest bound" of the set in the set.

[Jim Burns]

Post by Jim Burns

Post by Jim Burns
In arithmetic,
NUF(eps) ∉ ℕ

In arithmetic
∀x ∈ (0, 1]: y < x ==> y =< 0,
i.e., y is not positive.

In arithmetic,
∀eps > 0: eps > 0
i.e., eps is positive.

There is an implicit ∀y

I don't know which you intend:

(i)
(∀x ∈ (0,1]: y < x) ⇒ y ≤ 0
true

if y is a lower bound of (0,1]
then y ≤ glb(0,1] = 0

0 ∉ (0,1]

(ii)
∀x ∈ (0,1]: (y < x ⇒ y ≤ 0)
false

∀y: ∀x ∈ (0,1]: (y < x ⇒ y ≤ 0)
if and only if
¬∃y: ∃x ∈ (0,1]: (y < x ∧ 0 < y)
if and only if
¬∃x ∈ (0,1]: ∃y ∈ (0,x)
false
Instead,
∃x ∈ (0,1]: ∃y ∈ (0,x)

x ∈ ⋃{ (eps,1] | eps ∈ (0,1] }
if and only if
x ∈ (0,1]


∀eps > 0: NUF(eps) ∉ ℕ
true

Post by Jim Burns
Regards, WM

[WM]

Post by Jim Burns
(i)
(∀x ∈ (0,1]: y < x) ⇒ y ≤ 0
true

Yes, that is correct.

Regards, WM

[Richard Damon]

Post by WM

(i)
(∀x ∈ (0,1]: y < x)  ⇒  y ≤ 0
true

Yes, that is correct.
Regards, WM

Except we show it isn't for y = x/2

[Jim Burns]

Post by Richard Damon

Post by WM

(i)
(∀x ∈ (0,1]: y < x)  ⇒  y ≤ 0
true

Yes, that is correct.

Except we show it isn't for y = x/2

Well, for y = x/2, the antecedent is false,
thus the implication is true.
But WM probably isn't thinking of that.


(∀x ∈ (0,1]: y < x) ⇒ y ≤ 0
if and only if
0 < y ⇒ (∃x ∈ (0,1]: x ≤ y)

[WM]

Post by Fritz Feldhase

Post by WM
For the less-than relation there is no quantifier magic.

Was immer Deine "quantifier magic" auch sein soll; aber die (richtige)
An e IN: Em e IN: n < m (true)
Em e IN: An e IN: n < m (false)

Both are true, but the average person cannot look into the infinite.
Therefore I have invented the simpler example with unit fractions.
If for every x ∈ (0, 1] there are ℵ smaller unit fractions, then ℵ
unit fractions are smaller than every x ∈ (0, 1] and lie left-hand side
of the whole interval. Otherwise a first one would appear within the
interval.

Please note: Without ℵ unit fractions left-hand side of (0, 1], the
above true statement cannot be true.

Regards, WM

[Fritz Feldhase]

Post by WM

Post by Fritz Feldhase
An e IN: Em e IN: n < m (true)
Em e IN: An e IN: n < m (false)

Both are true,

Nope. "Em e IN: An e IN: n < m" is FALSE, otherwise there would have to be a natural number which is bigger than itself.

Post by WM
but the average person cannot look into the infinite.

Yeah whatever, Mückenheim.

[Ross Finlayson]

Post by Fritz Feldhase

Post by WM

Post by Fritz Feldhase
An e IN: Em e IN: n < m (true)
Em e IN: An e IN: n < m (false)

Both are true,

Nope. "Em e IN: An e IN: n < m" is FALSE, otherwise there would have to be a natural number which is bigger than itself.

Post by WM
but the average person cannot look into the infinite.

Yeah whatever, Mückenheim.

Yeah, that's wrong and Calvinistic, and rejected,
Duns Scotus and Spinoza give us a mathematical infinity,
that not only we can know, but according to Zeno, must.

That's not the same as "Absolute Infinity" and "the G-dhead",
which anyways are two things, that are named concepts with
universals in their definition.

Go away MW, Hodges won't be adding you to his hopeless.
An editor recalls some papers, ....

[Chris M. Thomasson]

Post by WM

Post by Fritz Feldhase

Post by WM
For the less-than relation there is no quantifier magic.

Was immer Deine "quantifier magic" auch sein soll; aber die (richtige)
An e IN: Em e IN: n < m (true)
Em e IN: An e IN: n < m (false)

Both are true, but the average person cannot look into the infinite.

You are the ultra finitist here... Pot Kettle?

Post by WM
Therefore I have invented the simpler example with unit fractions.
If for every x ∈ (0, 1] there are ℵ smaller unit fractions, then ℵ unit
fractions are smaller than every x ∈ (0, 1] and lie left-hand side of
the whole interval. Otherwise a first one would appear within the interval.
Please note: Without ℵ unit fractions left-hand side of (0, 1], the
above true statement cannot be true.
Regards, WM

[WM]

Post by WM

Post by Fritz Feldhase
An e IN: Em e IN: n < m (true)
Em e IN: An e IN: n < m (false)

Both are true,

Nope.

Yes, a mistake. Both are wrong. But the first statement is true for
*definable* numbers.

Regards, WM

[Richard Damon]

Post by Jim Burns

Post by Richard Damon

Post by WM

(i)
(∀x ∈ (0,1]: y < x)  ⇒  y ≤ 0
true

Yes, that is correct.

Except we show it isn't for y = x/2

Well, for y = x/2, the antecedent is false,
thus the implication is true.
But WM probably isn't thinking of that.

What part of the antecedent (∀x ∈ (0,1]: y < x) is false?

x is such that x ∈ (0,1], and y is such that y < x

Post by Jim Burns
(∀x ∈ (0,1]: y < x)  ⇒  y ≤ 0
 if and only if
0 < y  ⇒  (∃x ∈ (0,1]: x ≤ y)

[Jim Burns]

Post by Richard Damon

Post by Jim Burns

Post by Richard Damon

Post by WM

(i)
(∀x ∈ (0,1]: y < x)  ⇒  y ≤ 0
true

Yes, that is correct.

Except we show it isn't for y = x/2

Well, for y = x/2, the antecedent is false,
thus the implication is true.
But WM probably isn't thinking of that.

What part of the antecedent
(∀x ∈ (0,1]: y < x) is false?

Excuse me, I was thinking of something else.

As I'm sure you know, if
x ∈ (0,1] ⇒ y < x
is false for any value of x,
then all of
(∀x ∈ (0,1]: y < x)
is false,
and all of
(∀x ∈ (0,1]: y < x) ⇒ y ≤ 0
is true.

For y > 0 and x = min{y/2,1}
x ∈ (0,1] ⇒ y < x
is false
(∀x ∈ (0,1]: y < x)
is false, and
(∀x ∈ (0,1]: y < x) ⇒ y ≤ 0
is true.

On the other hand,
for y ≤ 0
(∀x ∈ (0,1]: y < x) ⇒ y ≤ 0
is true,
whatever the case is for
∀x ∈ (0,1]: y < x

In sum,
(∀x ∈ (0,1]: y < x) ⇒ y ≤ 0
is true, but
its truth doesn't tell us about y


I confess that I don't know
what significance that formula has to WM.

It's possible that, for WM,
talking about
(∀x ∈ (0,1]: y < x) ⇒ y ≤ 0
replaces talking about
∀eps > 0: NUF(eps) ∉ ℕ

[WM]

Post by Jim Burns
I confess that I don't know
what significance that formula has to WM.

If ∀x ∈ (0, 1]: NUF(x) = ℵo was true, it would prove negative unit
fractions y.

Regards, WM

[Jim Burns]

Post by WM

Post by Jim Burns
In sum,
(∀x ∈ (0,1]: y < x) ⇒ y ≤ 0
is true, but
its truth doesn't tell us about y
I confess that I don't know
what significance that formula has to WM.

If
∀x ∈ (0, 1]: NUF(x) = ℵo
was true,
it would prove negative unit fractions y.

How does
(∀x ∈ (0,1]: y < x) ⇒ y ≤ 0
and
∀x ∈ (0,1]: NUF(x) = ℵ₀
prove negative unit fractions y?

----
In arithmetic,
∀k <ℵ₀: ∀j <ℵ₀: j+k <ℵ₀

∀k <ℵ₀: ∀i <ℵ₀: i < |{j+k | j <ℵ₀}|

∀k <ℵ₀: ¬∃i <ℵ₀: i = |{j+k | j <ℵ₀}|

∀k <ℵ₀: ¬( |{j+k | j <ℵ₀}| <ℵ₀ )

----
In arithmetic,
∀k ∈ℕ₁: ∀j ∈ℕ: 0 < ⅟(j+k) ≤ 1

∀k ∈ℕ₁: {⅟(j+k) | j ∈ℕ} ⊆ (0,1] ∧
∀i ∈ℕ: i < |{⅟(j+k) | j ∈ℕ}|

∀k ∈ℕ₁: {⅟(j+k) | j ∈ℕ} ⊆ (0,1] ∧
¬∃i ∈ℕ: i = |{⅟(j+k) | j ∈ℕ}|

∀k ∈ℕ₁: {⅟(j+k) | j ∈ℕ} ⊆ (0,1] ∧
¬( |{⅟(j+k) | j ∈ℕ}| < |ℕ| )

At what formula, if any, have we stopped
using arithmetic?

Keep in mind that
_those formulas_ aren't dark.
You can see them on your screen.

[WM]

Post by Jim Burns

Post by WM

Post by Jim Burns
In sum,
(∀x ∈ (0,1]: y < x) ⇒ y ≤ 0
is true, but
its truth doesn't tell us about y
I confess that I don't know
what significance that formula has to WM.

If
∀x ∈ (0, 1]: NUF(x) = ℵo
was true,
it would prove negative unit fractions y.

How does
(∀x ∈ (0,1]: y < x) ⇒ y ≤ 0
and
∀x ∈ (0,1]: NUF(x) = ℵ₀
prove negative unit fractions y?

For every point x > 0 there are ℵ smaller unit fractions.
==>
There exists no point x > 0 without ℵ smaller unit fractions.
==>
There exist ℵ negative unit fractions.

Regards, WM

[Jim Burns]

Post by WM

Post by Jim Burns

Post by WM

Post by Jim Burns
In sum,
(∀x ∈ (0,1]: y < x)  ⇒  y ≤ 0
is true, but
its truth doesn't tell us about y
I confess that I don't know
what significance that formula has to WM.

If
∀x ∈ (0, 1]: NUF(x) = ℵo
was true,
it would prove negative unit fractions y.

How does
(∀x ∈ (0,1]: y < x)  ⇒  y ≤ 0
and
∀x ∈ (0,1]: NUF(x) = ℵ₀
prove negative unit fractions y?

For every point x > 0
there are ℵ smaller unit fractions.
==>
There exists no point x > 0 without
ℵ smaller unit fractions.
==>

https://sneltraining.nl/then-a-miracle-occurs/

"then a miracle occurs"

"I think you should be more explicit
here in step two"

Post by WM
There exist ℵ negative unit fractions.

An ordinal after each final ordinal
is not any of the final ordinals.
Thus, it is non.final.

A non.final ordinal, by definition,
is followed by
an ordinal with the same cardinality.

[WM]

Post by Richard Damon

Post by Jim Burns

Post by Richard Damon

Post by WM

(i)
(∀x ∈ (0,1]: y < x)  ⇒  y ≤ 0
true

Yes, that is correct.

Except we show it isn't for y = x/2

Well, for y = x/2, the antecedent is false,
thus the implication is true.
But WM probably isn't thinking of that.

What part of the antecedent (∀x ∈ (0,1]: y < x) is false?
x is such that x ∈ (0,1], and y is such that y < x

Only if y is less than all x ∈ (0,1], the implication holds. If the
antecedent is violated, the implication is true nevertheless. But that is
irrelevant.

Post by Richard Damon

Post by Jim Burns
(∀x ∈ (0,1]: y < x)  ⇒  y ≤ 0

Regards, WM

[Chris M. Thomasson]

Post by WM

Post by Jim Burns
Since you (WM) have decided that
you are talking about _not.arithmetic_
this might be an especially apt time
for your students to remember that
a claim about _not.arithmetic_
even if it were _true_
doesn't contradict a claim about _arithmetic_

That should not hinder an inquisitive student to learn that arithmetic
does not cover the domain of the smallest ℵo unit fractions and of the
largest ℵo natural numbers.

You need to fired immediately.

[Richard Damon]

Post by WM

Post by Jim Burns
Since you (WM) have decided that
you are talking about _not.arithmetic_
this might be an especially apt time
for your students to remember that
a claim about _not.arithmetic_
even if it were _true_
doesn't contradict a claim about _arithmetic_

That should not hinder an inquisitive student to learn that arithmetic
does not cover the domain of the smallest ℵo unit fractions and of the
largest ℵo natural numbers.
Regards, WM

But what arithmetic didn't cover them?

[Chris M. Thomasson]

Post by Richard Damon

Post by WM

Post by Jim Burns
Since you (WM) have decided that
you are talking about _not.arithmetic_
this might be an especially apt time
for your students to remember that
a claim about _not.arithmetic_
even if it were _true_
doesn't contradict a claim about _arithmetic_

That should not hinder an inquisitive student to learn that arithmetic
does not cover the domain of the smallest ℵo unit fractions and of the
largest ℵo natural numbers.
Regards, WM

But what arithmetic didn't cover them?

Telling a student that there is a largest natural number, should create
a scenario in which something should be fired? Just wondering here...

[Richard Damon]

Post by WM

Post by Jim Burns

Post by WM

Post by Jim Burns

Post by WM
Just today I showed some proofs
to my students.

If you are claiming your students for
some form of peer.review, then
you are accepting pre-calculus students as
_peers_

Nonsense.
There are 8th semester informatics and
engineering studens.

Perhaps
the informatics and engineering students of
Wolfgang Mückenheim of Hochschule Augsburg
remember their arithmetic.
kᵢⱼ = i+(i+j-1)(i+j-2)/2

Every number defined like k does not belong to the domain covered by the
smallest
ℵo unit fractions or by the largest ℵo natural numbers.
Regards, WM

Which of those largest natural numbers can k not get to?

What is the boundry that can not be passed?

[Ross Finlayson]

Post by Richard Damon

Post by WM

Post by Jim Burns

Post by WM

Post by Jim Burns

Post by WM
Just today I showed some proofs
to my students.

If you are claiming your students for
some form of peer.review, then
you are accepting pre-calculus students as
_peers_

Nonsense.
There are 8th semester informatics and
engineering studens.

Perhaps
the informatics and engineering students of
Wolfgang Mückenheim of Hochschule Augsburg
remember their arithmetic.
kᵢⱼ = i+(i+j-1)(i+j-2)/2

Every number defined like k does not belong to the domain covered by the
smallest
ℵo unit fractions or by the largest ℵo natural numbers.
Regards, WM

Which of those largest natural numbers can k not get to?
What is the boundry that can not be passed?

If you could pause on your brick-batting of MW a moment,
we're discussing foundations of arithmetic and algebra together.

Yet

[WM]

Post by Richard Damon
Which of those largest natural numbers can k not get to?

If I could name it I had made it v isible.

Post by Richard Damon
What is the boundry that can not be passed?

That is the difficult point: There is no fixed threshold. Most can't
comprehend it. Potential infinity!

Regards, WM

[Richard Damon]

Post by WM

Post by Richard Damon
Which of those largest natural numbers can k not get to?

If I could name it I had made it v isible.

Post by Richard Damon
What is the boundry that can not be passed?

That is the difficult point: There is no fixed threshold. Most can't
comprehend it. Potential infinity!
Regards, WM

So, you have two distinct sets with no boundry between them?

What makes them different?

If there isn't a line that keeps the describable numbers out of your
dark numbers, then aren't all your dark numbers describable?

If there is a line, then there must be a highest describable number, so
you can give it.

[Ross Finlayson]

Post by Richard Damon

Post by WM

Post by Richard Damon
Which of those largest natural numbers can k not get to?

If I could name it I had made it v isible.

Post by Richard Damon
What is the boundry that can not be passed?

That is the difficult point: There is no fixed threshold. Most can't
comprehend it. Potential infinity!
Regards, WM

So, you have two distinct sets with no boundry between them?
What makes them different?
If there isn't a line that keeps the describable numbers out of your
dark numbers, then aren't all your dark numbers describable?
If there is a line, then there must be a highest describable number, so
you can give it.

Sorites and the Heap.

There are no standard models of integers.

There are fragments, there are extensions, the ordinary inductive set's a non-logical constant.

There are multiple models of integers.

[Ross Finlayson]

p(a) v p(b) |- p(a) v p(b)
Whats the "corresponding" model?
I wonder whether your ass already envies your mouth,
for the amount of shit that comes out of it.

and that any proof has a corresponding model, and vice-versa.

You mean "TND: the domain of the universe of binary propositions?"

It's pretty simple that anything in proof theory has a model in model theory.

It's pretty simple that constructions of inferential relations are first-class,
objects in a model theory of a theory, the theory of model theory, its model.

To relate "first-class" and "first-order", is about simple representations of
logical constructions as logical constructions.


Heh, you wonder.


Seven deadly sins: ....

[Jim Burns]

On Sunday, January 14, 2024

[...]

Here are some things
you should be familiar with,
if you talking about models,
structures that embody all relations,
of integers,
here the natural or non-negative integers,
for that model-theory and proof-theory are
the same thing,
in terms of mathematical proofs,
insofar as that
a model of all relation and structurally,
is a proof,
and that
any proof has a corresponding model,
and vice-versa.

As I understand it,
the semantic (model.theory) and
the syntactic (proof.theory) points of view
are two sides of the same coin.

There are some very nice proofs showing
how the two are related, but
they aren't exactly the same thing.

For a certain theory (syntax),
there might or might not exist
a structure which it describes (semantics).
A structure which a theory describes
is a model of the theory.

Provably,
if no contradiction follows from a theory,
then a model of it exists.

The proof constructs (shows exists) a model from
the objects of the language of the theory.
That's why I note that, even if a theory has a model,
it still might not have the model you're thinking of.

Every structure that satisfies a theory is
a model of the theory.

Perhaps significantly different structures
satisfy the same theory (are models).

Provably,
if each model (semantics) which satisfies a theory
satisfies an additional formula,
then a proof (syntax) exists of that formula
starting from the theory (syntax).

----
Because there are formally undecidable formulas
in the natural numbers,
if any model exists,
then more than one model exists.

Also,
if any inductive set exists,
then a unique minimal inductive set exists,
subset to each inductive set, and
containing only each countable.to number.
By any other word, a standard model.

tl,;dr
The natural numbers either
are complete nonsense, top to bottom, or
have a standard model and a non.standard model.

[Ross Finlayson]

Post by Jim Burns

On Sunday, January 14, 2024

[...]

Here are some things
you should be familiar with,
if you talking about models,
structures that embody all relations,
of integers,
here the natural or non-negative integers,
for that model-theory and proof-theory are
the same thing,
in terms of mathematical proofs,
insofar as that
a model of all relation and structurally,
is a proof,
and that
any proof has a corresponding model,
and vice-versa.

As I understand it,
the semantic (model.theory) and
the syntactic (proof.theory) points of view
are two sides of the same coin.
There are some very nice proofs showing
how the two are related, but
they aren't exactly the same thing.
For a certain theory (syntax),
there might or might not exist
a structure which it describes (semantics).
A structure which a theory describes
is a model of the theory.
Provably,
if no contradiction follows from a theory,
then a model of it exists.
The proof constructs (shows exists) a model from
the objects of the language of the theory.
That's why I note that, even if a theory has a model,
it still might not have the model you're thinking of.
Every structure that satisfies a theory is
a model of the theory.
Perhaps significantly different structures
satisfy the same theory (are models).
Provably,
if each model (semantics) which satisfies a theory
satisfies an additional formula,
then a proof (syntax) exists of that formula
starting from the theory (syntax).
----
Because there are formally undecidable formulas
in the natural numbers,
if any model exists,
then more than one model exists.
Also,
if any inductive set exists,
then a unique minimal inductive set exists,
subset to each inductive set, and
containing only each countable.to number.
By any other word, a standard model.
tl,;dr
The natural numbers either
are complete nonsense, top to bottom, or
have a standard model and a non.standard model.

New RECUSITY lines...


RECUSITY:
FFAST



PRIDE


it's personal....




If you can read this,
look behind you.

I don't have eyes in the back of my head.




If wi-fi means wired fidelity,
I'm pretty sure it's cheating.



It's RECUSITY,
wear it out.



That model theory and proof theory are equi-interpretable,
and often thoroughly intermixed, pretty much separates
the men from the toys.

[Ross Finlayson]

Post by Ross Finlayson

Post by Jim Burns

On Sunday, January 14, 2024

[...]

Here are some things
you should be familiar with,
if you talking about models,
structures that embody all relations,
of integers,
here the natural or non-negative integers,
for that model-theory and proof-theory are
the same thing,
in terms of mathematical proofs,
insofar as that
a model of all relation and structurally,
is a proof,
and that
any proof has a corresponding model,
and vice-versa.

As I understand it,
the semantic (model.theory) and
the syntactic (proof.theory) points of view
are two sides of the same coin.
There are some very nice proofs showing
how the two are related, but
they aren't exactly the same thing.
For a certain theory (syntax),
there might or might not exist
a structure which it describes (semantics).
A structure which a theory describes
is a model of the theory.
Provably,
if no contradiction follows from a theory,
then a model of it exists.
The proof constructs (shows exists) a model from
the objects of the language of the theory.
That's why I note that, even if a theory has a model,
it still might not have the model you're thinking of.
Every structure that satisfies a theory is
a model of the theory.
Perhaps significantly different structures
satisfy the same theory (are models).
Provably,
if each model (semantics) which satisfies a theory
satisfies an additional formula,
then a proof (syntax) exists of that formula
starting from the theory (syntax).
----
Because there are formally undecidable formulas
in the natural numbers,
if any model exists,
then more than one model exists.
Also,
if any inductive set exists,
then a unique minimal inductive set exists,
subset to each inductive set, and
containing only each countable.to number.
By any other word, a standard model.
tl,;dr
The natural numbers either
are complete nonsense, top to bottom, or
have a standard model and a non.standard model.

New RECUSITY lines...
FFAST
PRIDE
it's personal....
If you can read this,
look behind you.
I don't have eyes in the back of my head.
If wi-fi means wired fidelity,
I'm pretty sure it's cheating.
It's RECUSITY,
wear it out.
That model theory and proof theory are equi-interpretable,
and often thoroughly intermixed, pretty much separates
the men from the toys.

Always a favorite:


I CAN READ

writing on the wall


Now with more THE.



James, your cogent response deserves a more thorough
and appreciative reply, so I hope that you'll find some
further in this thread, and I'll make some too.


Thanks for saving us all.

[Ross Finlayson]

Post by Ross Finlayson

Post by Ross Finlayson

Post by Jim Burns

On Sunday, January 14, 2024

[...]

Here are some things
you should be familiar with,
if you talking about models,
structures that embody all relations,
of integers,
here the natural or non-negative integers,
for that model-theory and proof-theory are
the same thing,
in terms of mathematical proofs,
insofar as that
a model of all relation and structurally,
is a proof,
and that
any proof has a corresponding model,
and vice-versa.

As I understand it,
the semantic (model.theory) and
the syntactic (proof.theory) points of view
are two sides of the same coin.
There are some very nice proofs showing
how the two are related, but
they aren't exactly the same thing.
For a certain theory (syntax),
there might or might not exist
a structure which it describes (semantics).
A structure which a theory describes
is a model of the theory.
Provably,
if no contradiction follows from a theory,
then a model of it exists.
The proof constructs (shows exists) a model from
the objects of the language of the theory.
That's why I note that, even if a theory has a model,
it still might not have the model you're thinking of.
Every structure that satisfies a theory is
a model of the theory.
Perhaps significantly different structures
satisfy the same theory (are models).
Provably,
if each model (semantics) which satisfies a theory
satisfies an additional formula,
then a proof (syntax) exists of that formula
starting from the theory (syntax).
----
Because there are formally undecidable formulas
in the natural numbers,
if any model exists,
then more than one model exists.
Also,
if any inductive set exists,
then a unique minimal inductive set exists,
subset to each inductive set, and
containing only each countable.to number.
By any other word, a standard model.
tl,;dr
The natural numbers either
are complete nonsense, top to bottom, or
have a standard model and a non.standard model.

New RECUSITY lines...
FFAST
PRIDE
it's personal....
If you can read this,
look behind you.
I don't have eyes in the back of my head.
If wi-fi means wired fidelity,
I'm pretty sure it's cheating.
It's RECUSITY,
wear it out.
That model theory and proof theory are equi-interpretable,
and often thoroughly intermixed, pretty much separates
the men from the toys.

I CAN READ
writing on the wall
Now with more THE.
James, your cogent response deserves a more thorough
and appreciative reply, so I hope that you'll find some
further in this thread, and I'll make some too.
Thanks for saving us all.

Calvin on Calvinism



Somebody had to keep it running

when nobody else knew how it did



Geil. Sehr geil.



In proof theory, it's provable, that in classical logic, one must
always assert Empty, Infinity, and well-foundedness for the
inference of the existence of any set, for otherwise inference
may arrive at Russell's set via quantification.


So, in classical logic, ZF's axioms aren't independent.
Of course, these days we know that that's "quasi-modal" logic.
Yet, it's not the point here that "classical" logic isn't monotonic.



Yet, the idea that "there's isn't a standard model of the integers",
and, also, "there are multiple non-standard models of integers",
in set theory, and as well in number theory, has a lot going on
with "there's an infinite integer in some infinitudes of integers".

Largely, an unbounded fragment, is the model of usual infinite induction.
That's not even necessarily an infinity, just an unbounded exercise
of induction.


Then, a true infinity, makes for integers, about the number-theoretic
properties of its elements, and what one has. This gets into things
like "primes at infinity", "double primes at infinity", "triple primes at
infinity", the last one of those being 2, 3, 5, and "quadruple primes at
infinity", the first one of those being at infinity, and on.



"Mathematics: truer than initially thought"


Now the above already has a lot why "there aren't standard models of
integers and there are bounded and unbounded fragments and there
are non-standard extensions", there's plenty more to it, not that that's
not plenty, to it.



"Hey, classical quasi-modal logic isn't all bad. You might get lucky."

[Jim Burns]

On Tuesday, January 16, 2024

[...]

In proof theory, it's provable, that
in classical logic,

proof.theory ≠ set.theory
proof ≠ set

logic < (non.logical) set.theory
set.theory = logic+set.description

set.description = infinity∨¬infinity
set.description = foundation∨¬foundation

different.description ⟺ different.models

one must always assert
Empty, Infinity, and well-foundedness
for the inference of the existence of any set,
for otherwise inference may arrive at
Russell's set via quantification.

Russell's ∃{x|x∉x} ⟸ unrestricted.comprehension

s/comprehension/specification+replacement

¬∃{x|x∉x} ⟸ classical
On the other hand,
revision.theory.of.truth+more?

So,
in classical logic,
ZF's axioms aren't independent.

Did you (RF) intend to make
that abrupt change of topic there?

That's not how I'd use "so"
Cha​cun à son goût.

Because models exist for
Infinity
¬Infinity
Foundation
¬Foundation
Choice
¬Choice
a (hypothetical) proof of
necessary.Infinity
necessary.Foundation
necessary.Choice
would be wrong.

tl;dr
Infinity, Foundation, Choice are
independent.

Of course, these days we know that
that's "quasi-modal" logic.

Uhm?
You mean?
https://philpapers.org/rec/GOLQEO-2
Quasi-Modal Equivalence of Canonical Structures

Yet, it's not the point here that
"classical" logic isn't monotonic.

Good to hear that it's not the point..
Classical is monotonic.

Yet, the idea that
"there's isn't a standard model of the integers",
and, also,
"there are multiple non-standard models of integers",
in set theory, and as well in number theory,
has a lot going on with
"there's an infinite integer in
some infinitudes of integers".

"has a lot going on with" = ?


Consider a finite set,
each element in its place.

A place is removed,
and not all elements have places.
We say:
the set's cardinality is finite.


Consider only all finite cardinals,
the ones which
cannot fit in a place removed.

For each finite cardinal,
inserting another element makes
a set which won't fit in its places,
one with a different cardinality.
Because finite.

Insert more and it still won't fit.
Insert all the cardinals and it still won't fit.

For each finite cardinal,
the cardinality of only all finite cardinals
is not its cardinality.

<drum.roll>
the cardinality of only all finite cardinals
is not a finite cardinality.
It is an infinite cardinality.
<cymbal.crash>

Check:
It isn't any cardinality which
cannot fit in a place removed.
It should fit in a place removed.

0@[1] 1@[2] 2@[3] 3@[4] ...
And check.

Weird as it is to fit in a place removed,
it must be so, or else
all the finite cardinals aren't
all the finite cardinals.

[Ross Finlayson]

Post by Jim Burns

On Tuesday, January 16, 2024

[...]

In proof theory, it's provable, that
in classical logic,

proof.theory ≠ set.theory
proof ≠ set
logic < (non.logical) set.theory
set.theory = logic+set.description
set.description = infinity∨¬infinity
set.description = foundation∨¬foundation
different.description ⟺ different.models

one must always assert
Empty, Infinity, and well-foundedness
for the inference of the existence of any set,
for otherwise inference may arrive at
Russell's set via quantification.

Russell's ∃{x|x∉x} ⟸ unrestricted.comprehension
s/comprehension/specification+replacement
¬∃{x|x∉x} ⟸ classical
On the other hand,
revision.theory.of.truth+more?

So,
in classical logic,
ZF's axioms aren't independent.

Did you (RF) intend to make
that abrupt change of topic there?
That's not how I'd use "so"
Cha​cun à son goût.
Because models exist for
Infinity
¬Infinity
Foundation
¬Foundation
Choice
¬Choice
a (hypothetical) proof of
necessary.Infinity
necessary.Foundation
necessary.Choice
would be wrong.
tl;dr
Infinity, Foundation, Choice are
independent.

Of course, these days we know that
that's "quasi-modal" logic.

Uhm?
You mean?
https://philpapers.org/rec/GOLQEO-2
Quasi-Modal Equivalence of Canonical Structures

Yet, it's not the point here that
"classical" logic isn't monotonic.

Good to hear that it's not the point..
Classical is monotonic.

Yet, the idea that
"there's isn't a standard model of the integers",
and, also,
"there are multiple non-standard models of integers",
in set theory, and as well in number theory,
has a lot going on with
"there's an infinite integer in
some infinitudes of integers".

"has a lot going on with" = ?
Consider a finite set,
each element in its place.
A place is removed,
and not all elements have places.
the set's cardinality is finite.
Consider only all finite cardinals,
the ones which
cannot fit in a place removed.
For each finite cardinal,
inserting another element makes
a set which won't fit in its places,
one with a different cardinality.
Because finite.
Insert more and it still won't fit.
Insert all the cardinals and it still won't fit.
For each finite cardinal,
the cardinality of only all finite cardinals
is not its cardinality.
<drum.roll>
the cardinality of only all finite cardinals
is not a finite cardinality.
It is an infinite cardinality.
<cymbal.crash>
It isn't any cardinality which
cannot fit in a place removed.
It should fit in a place removed.
And check.
Weird as it is to fit in a place removed,
it must be so, or else
all the finite cardinals aren't
all the finite cardinals.

Again your reasoning shows you in a good light,
I'll look to it.

About the "so, ..., axioms of restriction aren't independent...",
it's the inference as of what is directly above, or, "no, not non sequitur".

It's pretty interesting and fun that model theory and proof theory
are equi-interpretable, intermixable, and not allowed to contradict each other.

Also free comprehension always exists.

[Ross Finlayson]

Post by Ross Finlayson

Post by Jim Burns

On Tuesday, January 16, 2024

[...]

In proof theory, it's provable, that
in classical logic,

proof.theory ≠ set.theory
proof ≠ set
logic < (non.logical) set.theory
set.theory = logic+set.description
set.description = infinity∨¬infinity
set.description = foundation∨¬foundation
different.description ⟺ different.models

one must always assert
Empty, Infinity, and well-foundedness
for the inference of the existence of any set,
for otherwise inference may arrive at
Russell's set via quantification.

Russell's ∃{x|x∉x} ⟸ unrestricted.comprehension
s/comprehension/specification+replacement
¬∃{x|x∉x} ⟸ classical
On the other hand,
revision.theory.of.truth+more?

So,
in classical logic,
ZF's axioms aren't independent.

Did you (RF) intend to make
that abrupt change of topic there?
That's not how I'd use "so"
Cha​cun à son goût.
Because models exist for
Infinity
¬Infinity
Foundation
¬Foundation
Choice
¬Choice
a (hypothetical) proof of
necessary.Infinity
necessary.Foundation
necessary.Choice
would be wrong.
tl;dr
Infinity, Foundation, Choice are
independent.

Of course, these days we know that
that's "quasi-modal" logic.

Uhm?
You mean?
https://philpapers.org/rec/GOLQEO-2
Quasi-Modal Equivalence of Canonical Structures

Yet, it's not the point here that
"classical" logic isn't monotonic.

Good to hear that it's not the point..
Classical is monotonic.

Yet, the idea that
"there's isn't a standard model of the integers",
and, also,
"there are multiple non-standard models of integers",
in set theory, and as well in number theory,
has a lot going on with
"there's an infinite integer in
some infinitudes of integers".

"has a lot going on with" = ?
Consider a finite set,
each element in its place.
A place is removed,
and not all elements have places.
the set's cardinality is finite.
Consider only all finite cardinals,
the ones which
cannot fit in a place removed.
For each finite cardinal,
inserting another element makes
a set which won't fit in its places,
one with a different cardinality.
Because finite.
Insert more and it still won't fit.
Insert all the cardinals and it still won't fit.
For each finite cardinal,
the cardinality of only all finite cardinals
is not its cardinality.
<drum.roll>
the cardinality of only all finite cardinals
is not a finite cardinality.
It is an infinite cardinality.
<cymbal.crash>
It isn't any cardinality which
cannot fit in a place removed.
It should fit in a place removed.
And check.
Weird as it is to fit in a place removed,
it must be so, or else
all the finite cardinals aren't
all the finite cardinals.

Again your reasoning shows you in a good light,
I'll look to it.
About the "so, ..., axioms of restriction aren't independent...",
it's the inference as of what is directly above, or, "no, not non sequitur".
It's pretty interesting and fun that model theory and proof theory
are equi-interpretable, intermixable, and not allowed to contradict each other.
Also free comprehension always exists.

Hi James,

I enjoyed this talk, it helps reflect quite a thorough theoretical approach
to continuity and about the objects of mathematics sort of generally.

There's lots of talk here about infinitesimals that don't exist. Now, maybe it will
help if we sort of delineate the classical notions, of what these things are.

The ordered field is Archimedean. It has the infinitely-many, the unbounded,
and, no infinite magnitudes. Now, its result that way, sort of has to include
how it's arrived at. Archimedes, first establishes there _are_ infinitely-many,
there are not found bounds, and, then that there are no infinite magnitudes.
This then is very Pythagorean, where, the ordered field of rationals, yet of
course has no ir-rational quantities, either. This then wouldn't suffice for
most today, of course, because we have a notion of a complete ordered field,
that's a model of a linear continuum as of our model of a linear curriculum.

Then, what seems most these people are ignorant, is Aristotle's continuum.
Now, Aristotle is most well-known for inductive inference after syllogism.
But, there's also has deductive inference, for Aristotle's reflections on Parmenides.
Here though, the notion of the Aristotle's continuum, is that the segment,
is finite, but is equi-partitioned with infinitely-many. This is the most simple
usual model of line-drawing or putting pencil to paper, drawing, and lifting it,
relating exactly this model in space according to any model in time, in exactly
the course-of-passage, in space through time.


So, it's something to be known that when someone introduces this sort of
notion, Aristotle vis-a-vis Archimedes, that there's plenty going on in their
considerations of the replete courtesy courtesy their simple relations of
the mental apparatus.

These days for example there are notions of "non-Archimedean fields". Now,
you might look at something like Groethendieck universes, which is in the
space of words for algebra, and entirely separate the notions of the space
of words, and the space of points. They end up sort of being at odds.

So, is the complete ordered field, after Eudoxus' construction of the rational
moduli and its extension about methods of exhaustion that the circle is no
different than the infinitely-sided regular polygon, in the infinite limit?
This is of course that methods of exhaustion or the infinite limit was a notion
long before Weierstrass helped bolt it down with a passel of delta-epsilonics.
So, Cauchy/Dedekind is just a patching of Eudoxus-to-Weierstrass into abstract
algebra in a space of words.

Is the complete ordered field, ..., Archimedean and Pythagorean, anymore?
The field of rationals is, yes, indeed, but, _there is more to it_, it's not so much
that "it's more than the sum of its parts" as "it IS the sum of its parts".

So, "iota-values", or "standard infinitesimals", "modern paleo-classical Aristotle's
constant monotone increasing vanishing values with a different lower and upper bound",
is ANCIENT and already long, long, long ago part of the canon.

Then, to make it modern again, is for a simple few results in function theory for line-reals,
and topology for signal-reals, resulting at least THREE of more replete definitions of
continuity and continuous domains, that exist in the universe of objects of mathematics.

Where nothing non-mathematical does.


Now I'm very much enjoying your introduced notation, and as well particularly
your attention to quantifier disambiguation, I generally frame it in this sort
of way: that the universal quantifier has various forms for-any/for-each/for-every/for-all,
that these are "four different for's", that in some conditions are same and others different.

This is about quite directly the Sorites and Heap, which otherwise you'll find
all these quite standard modern adherents to exoteric dogma tripping all over.

So, the transfer principles, that what's so for the elements is so for their collection,
you'll find a lot, it's the usual definition of ordinals in set theory.

Yet, in objects of mathematics, ordering theories in ordinals their sets,
and set theories in sets their ordinals, are two different models that result
"the same universe, all full of them", about ubiquitous ordinals and universal algebra.

I feel like, as a conscientious researcher in foundations, it demands a lot.
So, pretty much I'm a strong Platonist, a strong logical positivist,
have a strong mathematical universe and mathematical hypothesis,
then, I guess I'm Finlaysonist. Or, I'm looking all around and point at me.

Pretty strongly, ....

Finlaysonism: strong mathematical platonism, strong logical positivism,
strong mathematical universal hypothesism, sole and true foundations.


Model theory and proof

[Jim Burns]

On Wednesday, January 17, 2024

On Wednesday, January 17, 2024

[...]

[...]

Here though,
the notion of the Aristotle's continuum, is that
the segment, is finite, but
is equi-partitioned with infinitely-many.
This is the most simple usual model of line-drawing or
putting pencil to paper, drawing, and lifting it,
relating exactly this model in space according to
any model in time, in exactly the course-of-passage,
in space through time.

My vague sense of these issues is that
Zeno argued very effectively that
that simplest theory of the line is too simple.

Our current theory of the line,
a theory stress.tested beyond the dreams of Zeno,
is that, in topological terms,
the line is _one component_ that it can't be
partitioned into two non.empty open sets.

Without that being true,
there can be curves y=f(x) which are continuous at
each point of the line, but which, nonetheless,
jump over lines between <x1,y1> and <x2,y2>

Since jumping over isn't the sort of behavior
which the things we consider curves get up to,
it seems unavoidable for us to require
_one component_

One effect of there being only one component is
that any countable series of points must have
a split with no series.point between sides.
But there must be a point.between,
or there will be more than one component.

One component implies a point.between, but
a point.between which is not in the series.
And so on, for each series of points.
_Each_ series not holding all points
implies that mere infinitely.small partitions
aren't enough for a good theory of the line.

[WM]

There are no unit fractions smaller that ALL unit fractions

But there are ℵ unit fraction smaller than all you can name.

No, there are not. As I have shown, I can name any of them.

Then show it. Name a unit fraction that has not ℵ smaller ones.

No, the number of unit fractions between 0 and x will always be Alpha_0,
the measure of Countable infinity, as there will ALWAYS be that many
unit fractions between 0 and x (unless x is 0, then the number is 0)

The term is aleph_0, not alpha_0.
The mathematics is this: if x is less than every positive number, then x
is less than (0, oo). That is impossible for positive x, let alone for
ℵo unit fractions.

∀eps > 0 ∀x ∈ (eps, 1]: NUF(x) = ℵo

Right. There are always ℵo unit fractions below a positive number, so no
number exists where NUF(x) == 1

The statement says not below "a positive number" but below "every positive
number", but that means below (0, oo) and is nonsense.

Your logic is just insufficient to handle unbounded sets.

My logic is based on mathematics. Your claims are not logic but dogmas of
matheology in contradiction with mathematics.

Regards, WM

[Richard Damon]

Post by WM

There are no unit fractions smaller that ALL unit fractions

But there are ℵ unit fraction smaller than all you can name.

No, there are not. As I have shown, I can name any of them.

Then show it. Name a unit fraction that has not ℵ smaller ones.

So, you don't unddrstand English and just working with Strawman.

I never claimed there was a unit fraction that doesn't have aleph_0
smaller ones.

Post by WM

No, the number of unit fractions between 0 and x will always be
Alpha_0, the measure of Countable infinity, as there will ALWAYS be
that many unit fractions between 0 and x (unless x is 0, then the
number is 0)

The term is aleph_0, not alpha_0.
The mathematics is this: if x is less than every positive number, then x
is less than (0, oo). That is impossible for positive x, let alone for
ℵo unit fractions.

So, you just proved that there is no

Post by WM

∀eps > 0 ∀x ∈ (eps, 1]: NUF(x) = ℵo

Right. There are always ℵo unit fractions below a positive number, so
no number exists where NUF(x) == 1

The statement says not below "a positive number" but below "every
positive number", but that means below (0, oo) and is nonsense.

So, you agree that NUF(x) = ℵo for all positive x and is never a finite
value.

Since for all eps > 0 NUF(x) == ℵo, there is no eps for which NUF(x) is
smaller than that, not even a "dark" one.

Post by WM

Your logic is just insufficient to handle unbounded sets.

My logic is based on mathematics. Your claims are not logic but dogmas
of matheology in contradiction with mathematics.

Nope. YOUR claims are based on the dogma of WMism, not actual logic or
mathematics.

You can't get to actual fundamental statements that show your claims, so
you are just shown to be a dogmatist.

Post by WM
Regards, WM

[Fritz Feldhase]

Name a unit fraction that has not ℵ[0] smaller ones.

Why should he "name" something that doesn't exist, you psychotic asshole full of shit?!

_________________

Btw. It's possible to introduce a "naming schema" such that each and every unit fraction has a name:

If u is a unit fraction, then the symbol "I/I...I" where "I...I" consists of 1/u "I"s is the name of u.

For example, if u = 1/3, then "I/III" is its name. The other way round: "|/|||" has 3 occurences of the symbol "I" after the symbol "/". Hence it's the name of the unit fraction 1/3.

[Fritz Feldhase]

It is sufficient that an endless digit sequence [...] exists.

No. Even and endless digit sequence does not describe an irrational number.

It does, you silly idiot.

See: https://www.dpmms.cam.ac.uk/~wtg10/decimals.html

[Ross Finlayson]

Post by Fritz Feldhase

It is sufficient that an endless digit sequence [...] exists.

No. Even and endless digit sequence does not describe an irrational number.

It does, you silly idiot.
See: https://www.dpmms.cam.ac.uk/~wtg10/decimals.html

Where's Simon Stevin when you need him?

[Archimedes Plutonium]

Mueckenheim weighs Dr.Laura Covi,Dr.Andreas Dillmann,Dr. Stefan Dreizler for no-one at Gottingen can do a proper Water Electrolysis and actually weigh the hydrogen and oxygen. No, these fools only look at volume.



Mathin3D, Volney weighs Gottingen for they are too stupid to look beyond volume in Water Electrolysis and weigh the actual mass of hydrogen and oxygen to prove AP correct-- water is really H4O, not H2O.

Dr.Richard Schrock,NSF Dr.Panchanathan


Gottingen Uni,Dr. Sarah Köster,Dr. Reiner Kree,Dr. Matthias Krüger


Volney Physics failures..Gottingen Uni,Dr. Sarah Köster,Dr. Reiner Kree,Dr. Matthias Krüger, NSF Dr.Panchanathan,Alejandro Adem, Purdue Univ_France Cordova,

Jan Burse and Volney, why this eternal september spam cluttering up the newsgroup.

LIGAPEDIA by Belinda
PUTRITOTO SLOT by echa
PUTRITOTO by Putri


Why Volney?? Because they are so sloppy and slipshod in Physics experiment of Water Electrolysis, stopping and ceasing the experiment before weighing the mass of the hydrogen compared to mass of oxygen. Is it that they are stupid silly thinking volume and mass are the same. For AP needs to prove decisively, if Water is really H4O or H2O. And of course, this experiment would destroy the Standard Model-- that post-diction theory of physics that never gave a single prediction in all of its tenure.

And they even know that a weighing balance of Quartz Crystal MicroBalance has been around since the 1960s, what are they waiting for???

Or is it because they cannot admit the truth of math geometry that slant cut of cone is oval, not ellipse for you need the symmetry of slant cut of cylinder to yield a ellipse.


,
Volney
3
WM using AP's TEACHING TRUE MATHEMATICS to teach 13-14 year olds CALCULUS, those heading for Gottingen & Uni Berlin for AP reduced Calculus to its most simple form-- add or subtract 1 from exponent.
9:01 PM


Universitat Augsburg, Germany, rector Sabine Doering-Manteuffel
Math dept Ronald H.W.Hoppe, B. Schmidt, Sarah Friedrich, Stefan Grosskinsky, Friedrich Pukelsheim, Mirjam Dur, Ralf Werner.

Hochschule Augsburg, Wolfgang Mueckenheim

Eternal-September.org
Wolfgang M. Weyand
Berliner Strasse
Bad Homburg

Goethe Universitat Physics dept

Brigitta Wolff president

Jurgen Habermass
Horst Stocker
Gerd Binnig
Horst Ludwig Stormer  
Peter Grunberg

math
Alex Kuronya
Martin Moller
Jakob Stix
Annette Werner
Andreas Bernig
Esther Cabezas-Rivas
Hans Crauel
Thomas Gerstner
Bastian von Harrach
Thomas Mettler
Tobias Weth
Amin Coja-Oghlan
Raman Sanyal
Thorsten Theobald
Yury Person            

Gottingen Univ math

Metin Tolan

Dorothea Bahns, Laurent Bartholdi, Valentin Blomer, Jorg Brüdern, Stefan Halverscheid, Harald Andres Helfgott, Madeleine Jotz Lean, Ralf Meyer, Preda Mihailescu, Walther Dietrich Paravicini, Viktor Pidstrygach, Thomas Schick, Evelina Viada, Ingo Frank Witt, Chenchang Zhu

Gottingen Univ physics
Prof. Dr. Karsten Bahr
Prof. Dr. Peter Bloechl
Prof. Dr. Eberhard Bodenschatz
Prof. Laura Covi, PhD
Prof. Dr. Andreas Dillmann
Prof. Dr. Stefan Dreizler
Prof. Dr. Jörg Enderlein
Prof. Dr. Laurent Gizon
Prof. Dr. Ariane Frey
apl. Prof. Dr. Wolfgang Glatzel
Prof. Dr. Fabian Heidrich-Meisner
Prof. Dr. Hans Christian Hofsäss
Prof. Dr. Andreas Janshoff
Prof. Dr. Christian Jooß
Prof. Dr. Stefan Kehrein
Prof. Dr. Stefan Klumpp
Prof. Dr. Sarah Köster
Prof. Dr. Reiner Kree
Prof. Dr. Matthias Krüger
Prof. Dr. Stanley Lai
Prof. Dr. Stefan Mathias
apl. Prof. Dr. Vasile Mosneaga
Prof. Dr. Marcus Müller
Prof. Dr. Jens Niemeyer
apl. Prof. Dr. Astrid Pundt
Prof. Dr. Arnulf Quadt
apl. Prof. Dr. Karl-Henning Rehren
Prof. Dr. Ansgar Reiners
Prof. Dr. Angela Rizzi
Prof. Dr. Claus Ropers
Prof. Dr. Tim Salditt
Prof. Dr. Konrad Samwer
Prof. Dr. Christoph Schmidt
apl. Prof. Dr. Susanne Schneider
Prof. Dr. Steffen Schumann
Prof. Dr. Simone Techert
apl. Prof. Dr. Michael Seibt
Prof. Dr. Peter Sollich
Prof. Dr. Andreas Tilgner
Prof. Cynthia A. Volkert
Prof. Dr. Florentin Wörgötter
Prof. Dr. Annette Zippelius

educ dept of Germany

Bettina Stark-Watzinger, Jens Brandenburg, Thomas Sattelberger, Kornelia Haugg, Judith Pirscher

Apparently Kibo realized he was a science failure when he could not even do a proper percentage. But then one has to wonder how much he paid to bribe Rensselaer to graduate from the school in engineering unable to do a percentage properly???? For I certainly would not hire a engineer who cannot even do proper percentage.

Silly boy, that's off by more than 12.6 MeV, or 12% of the mass of a muon.
Hardly "exactly" 9 muons.
Or, 938.2720813/105.6583745 = 8.88024338572. A proton is about the mass
of 8.88 muons, not 9. About 12% short.

Why Volney?? Because they stop short of completing the Water Electrolysis Experiment by only looking at volume, when they are meant to weigh the mass of hydrogen versus oxygen?? Such shoddy minds in experimental physics and chemistry.

Rensselaer Polytechnic Institute Physics dept Dr.Martin Schmidt (ee), Dr.Ivar Giaever
Vincent Meunier, Ethan Brown, Glenn Ciolek, Julian S. Georg, Joel T. Giedt, Yong Sung Kim, Gyorgy Korniss, Toh-Ming Lu, Charles Martin, Joseph Darryl Michael, Heidi Jo Newberg, Moussa N'Gom, Peter Persans, John Schroeder, Michael Shur, Shawn-Yu Lin, Humberto Terrones, Gwo Ching Wang, Morris A Washington, Esther A. Wertz, Christian M. Wetzel, Ingrid Wilke, Shengbai Zhang

Rensselaer math department
Donald Schwendeman, Jeffrey Banks, Kristin Bennett, Mohamed Boudjelkha, Joseph Ecker, William Henshaw, Isom Herron, Mark H Holmes, David Isaacson, Elizabeth Kam, Ashwani Kapila, Maya Kiehl, Gregor Kovacic, Peter Kramer, Gina Kucinski, Rongjie Lai, Fengyan Li, Chjan Lim, Yuri V Lvov, Harry McLaughlin, John E. Mitchell, Bruce Piper, David A Schmidt, Daniel Stevenson, Yangyang Xu, Bulent Yener, Donald Drew, William Siegmann

Physics minnow

What warning is that Kibo Parry failure of science-- warning that insane persons like Kibo Parry Moroney Volney spends their entire life in a hate-mill, never doing anything in science itself. And paid to stalk hate spew

Kibo Parry Moroney-Volney blowing his cover with the CIA in 1997

Re: Archimedes Vanadium, America's most beloved poster

---quoting Wikipedia ---

Controversy
Many government and university installations blocked, threatened to block, or attempted to shut-down The World's Internet connection until Software Tool & Die was eventually granted permission by the National Science Foundation to provide public Internet access on "an experimental basis."

--- end quote ---

NATIONAL SCIENCE FOUNDATION
Dr. Panchanathan , present day
NSF Dr. Panchanathan, F. Fleming Crim, Dorothy E Aronson, Brian Stone, James S Ulvestad (math), Rebecca Lynn Keiser, Vernon D. Ross, Lloyd Whitman, John J. Veysey (physics), Scott Stanley
France Anne Cordova
Subra Suresh (bioengineer)
Arden Lee Bement Jr. (nuclear engineering)
Rita R. Colwell (microbiology)
Neal Francis Lane
John Howard Gibbons 1993
Barry Shein, kibo parry std world
Jim Frost, Joe "Spike" Ilacqua
Canada-- NSERC , Alejandro Adem (math) , Navdeep Bains, Francois-Philippe Champagne

News starting to come in that AP's Water Electrolysis Experiment proves the true formula of Water is H4O, not H2O is starting to come in.

Aug 30, 2023, 10:19:20 PM
to Plutonium Atom Universe
News starting to come in that AP's Water Electrolysis Experiment proves the true formula of Water is H4O, not H2O is starting to come in.
I received a letter today of Experiment results on Water Electrolysis of weighing the hydrogen test tube versus oxygen test tube and the result is 1/4 atomic mass units of Hydrogen compared to Oxygen.
The researcher weighing 1600 micrograms of hydrogen, using a Eisco-Brownlee-Water-Electrolysis Apparatus.
Using sulfuric acid as electrolyte on ultra pure water. Using low voltage DC of 1.5 volts, 1 amp.
I am not surprised that news of the true formula of Water is H4O comes so quickly. For not much in science is more important than knowing the truth of Water. And this means the start of the complete downfall and throwing out the sick Standard Model of Physics, for it is such an insane theory that it cannot get passed the idea of its subatomic particles as stick and ball, with no job, no function, no task. The Standard Model of Physics, is crazy insane physics for it is all postdiction, never prediction. The idea that the hydrogen atom is H2 not H, is because of the prediction of Atom Totality Theory where a atom is a proton torus with muon inside doing the Faraday law and all atoms require at least 1 capacitor. That means the one proton in H2 serves as a neutron to the other proton, storaging the electricity produced by the other proton.
The true Hydrogen Atom is H2 for all atoms need at least one capacitor, and one of the protons in H2 serves as a neutron.
Sad that chemistry and physics throughout the 20th century were too stupid to actually weigh the mass of hydrogen and oxygen in electrolysis, no, the ignorant fools stopped at looking when they saw the volume of hydrogen was twice that of oxygen. A real scientist is not that shoddy and slipshod ignorant, the real scientist then proceeds with -- let us weigh the hydrogen test tube mass versus the oxygen test tube mass.
Thanks for the news!!!!!
AP
News starting to come in that AP's Water Electrolysis Experiment proves the true formula of Water is H4O, not H2O is starting to come in.
There is another experiment that achieves the same result that Water is truly H4O and not H2O, but I suspect this second method is hugely fraught with difficulty.
The prediction of H4O comes from the Physics idea that a Atom is composed, all atoms mind you, is composed of a proton torus with muon/s inside going round and round thrusting through the torus in the Faraday law and producing electricity. So that when you have Hydrogen without a neutron, there is no way to collect the electricity produced by the Faraday law. Think of it as a automobile engine, you cannot have a engine if there is no crank shaft to collect the energy from the thrusting piston inside the crankcase.
Same thing with an Atom, it needs 3 parts-- muon as bar magnet, proton as torus of coils, and a capacitor to storage the produced electricity. If one of those parts is missing, the entity is a Subatomic particle and not a atom.
So, when we have Hydrogen as a proton with muon inside, it is not a Atom, until it has a neutron, or, has another proton of hydrogen H2, then it is a Atom.
So that H2 is not a molecule but a Atom. H alone is a subatomic particle.
Much harder than Water Electrolysis.
We need to get two identical containers.
We need to be able to make pure heavy-water with deuterium. Deuterium is proton + neutron as hydrogen. Proton + proton is H2 as hydrogen.
So we have two identical containers and we fill one with pure heavy water, deuterium water.
We have the second container and we fill it with pure (light) water.
We now weigh both of them.
If AP is correct, that water is really H4O and not H2O, then both containers should weigh almost the same. Only a tiny fraction difference because the neutron is known to be 940MeV versus proton in Old Physics as 938MeV a tiny difference of 2MeV, but we realize we have a huge number of water molecules in the two identical containers.
If water is truly H4O, the weights should be almost the same. If water is H2O, then there is a **large difference** in weights.
But the Water Electrolysis experiment is much easier to conduct and get results.
And, there is the biological processes that apparently cannot distinguish between heavy water and that of regular normal water.
Deuterium Water is the same in biology as is normal regular water. This means that water must be H4O, due to biology as proof.
Deuterium Water in atomic mass units is 16 for the oxygen and 4 for the deuterium.
Regular normal Water in atomic mass units is 16 for the oxygen and 4 for the 4 protons in H4O.
Old Physics and Old Chemistry had regular water as H2O in atomic mass units of 16 oxygen and 2 hydrogen for 2 protons.
If biology functions whether heavy water or normal water all the same, then water itself must be H4O.
Now, there maybe some animal or plant that can separate out heavy water from H4O water???
Searching the literature today for where biology needs as essential deuterium water. And not too surprised that it is a essential requirement in metabolism. In fact one web site listed the need for deuterium more than the need of many minerals and vitamins.
Now tonight I came up with two new exciting experiments to verify that Water is truly H4O and not H2O.
H4O is 4 protons with muons inside the 840MeV proton toruses.
Deuterium water is DOD. And the difference between D2O and H4O is merely the difference of 4MeV for as last reported, neutron = 940MeV and proton (with muon inside) is 938MeV, a difference of 2MeV but for water is 2+2 = 4MeV.
So these two new experiments take advantage of the fact that what we think is normal regular water is actually very close to heavy water of D2O, with only a 4MeV difference.
EXPERIMENT #3 Water layers in still pond of D2O mixed with H4O (what we thought was H2O.
So in this experiment we get a clear glass container and mix H4O with D2O. If Old Physics is correct, the heavy water should sink rapidly in the container while the light water floats to the top rapidly. And we have some sort of beam of photons that can distinguish D2O from H4O (thought of as H2O. We obtain pure D2O and pure H4O each filling 1/2 of the container. We stir and mix them. And then we observe with the EM beam for separation. If the light water is truly H4O, it takes a long time for the D2O to be on bottom and H4O on top. We measure the time of a settled container and determine this time from the theoretical 4MeV difference should take a long time, whereas if Old Physics is correct, the separation would be almost instantly and quick time.
EXPERIMENT #4 also plays on this minor difference of 4MeV. We devise a sort of squirt gun for D2O and a identical squirt gun for H4O (what we call H2O). We put pure D2O in one squirt gun and the H40 or light water in the other squirt gun. Both guns forcing the water a certain distance.
If AP is correct that light water is really H4O and not H2O as we squirt both guns, where the water lands should be almost the same distance considering H4O is only 4MeV apart from D2O.
If Old Physics and Old Chemistry is correct, then D2O water is 940 + 940 = 1880MeV apart from light water of H2O, and H4O is only 4MeV apart.
So where the squirt gun lands the D2O is a very much shorter distance than a H2O land, but a H4 land distance is nearly the same as the D2O land.
These two experiments are very exciting and would be a very nice confirming evidence to Water Electrolysis actual weighing the mass in atomic mass units.

Searching the literature today for where biology needs as essential deuterium water. And not too surprised that it is a essential requirement in metabolism. In fact one web site listed the need for deuterium more than the need of many minerals and vitamins.
Now tonight I came up with two new exciting experiments to verify that Water is truly H4O and not H2O.
H4O is 4 protons with muons inside the 840MeV proton toruses.
Deuterium water is DOD. And the difference between D2O and H4O is merely the difference of 4MeV for as last reported, neutron = 940MeV and proton (with muon inside) is 938MeV, a difference of 2MeV but for water is 2+2 = 4MeV.
So these two new experiments take advantage of the fact that what we think is normal regular water is actually very close to heavy water of D2O, with only a 4MeV difference.
EXPERIMENT #3 Water layers in still pond of D2O mixed with H4O (what we thought was H2O.
So in this experiment we get a clear glass container and mix H4O with D2O. If Old Physics is correct, the heavy water should sink rapidly in the container while the light water floats to the top rapidly. And we have some sort of beam of photons that can distinguish D2O from H4O (thought of as H2O. We obtain pure D2O and pure H4O each filling 1/2 of the container. We stir and mix them. And then we observe with the EM beam for separation. If the light water is truly H4O, it takes a long time for the D2O to be on bottom and H4O on top. We measure the time of a settled container and determine this time from the theoretical 4MeV difference should take a long time, whereas if Old Physics is correct, the separation would be almost instantly and quick time.

Apparently this Experiment is already done and called for-- There is Uniform Distribution of heavy water Deuterium Water in the Oceans, Lakes, Ponds, Streams and Rivers. Heavy Water is not layered in the oceans or lakes or ponds or streams or rivers. Uniformity means that the difference between D2O and H4O is so slight of a difference (only 4MeV, compared to 1880MeV for H2O, that Brownian motion keeps the D2O and H4O in a Uniform Distribution in all bodies of water. I was going through the research literature today and find that scientists discover Uniformity of Distribution of deuterium water. This thus closes the case on Water, for uniformity of distribution of D2O implies that Water is itself H4O and not H2O.

My 250th published book.

TEACHING TRUE CHEMISTRY; H2 is the hydrogen Atom and water is H4O, not H2O// Chemistry
by Archimedes Plutonium (Author) (Amazon's Kindle)
Prologue: This textbook is 1/2 research history and 1/2 factual textbook combined as one textbook. For many of the experiments described here-in have not yet been performed, such as water is really H4O not H2O. Written in a style of history research with date-time markers, and fact telling. And there are no problem sets. This book is intended for 1st year college. Until I include problem sets and exercises, I leave it to the professor and instructor to provide such. And also, chemistry is hugely a laboratory science, even more so than physics, so a first year college student in the lab to test whether Water is really H4O and not H2O is mighty educational.
Preface: This is my 250th book of science, and the first of my textbooks on Teaching True Chemistry. I have completed the Teaching True Physics and the Teaching True Mathematics textbook series. But had not yet started on a Teaching True Chemistry textbook series. What got me started on this project is the fact that no chemistry textbook had the correct formula for water which is actually H4O and not H2O. Leaving the true formula for hydroxyl groups as H2O and not OH. But none of this is possible in Old Chemistry, Old Physics where they had do-nothing subatomic particles that sit around and do nothing or go whizzing around the outside of balls in a nucleus, in a mindless circling. Once every subatomic particle has a job, task, function, then water cannot be H2O but rather H4O. And a hydrogen atom cannot be H alone but is actually H2. H2 is not a molecule of hydrogen but a full fledged Atom, a single atom of hydrogen.
Cover Picture: Sorry for the crude sketch work but chemistry and physics students are going to have to learn to make such sketches in a minute or less. Just as they make Lewis diagrams or ball & stick diagrams. My 4-5 minute sketch-work of the Water molecule H4O plus the subatomic particle H, and the hydrogen atom H2. Showing how one H is a proton torus with muon inside (blue color) doing the Faraday law. Protons are toruses with many windings. Protons are the coils in Faraday law while muons are the bar magnets. Neutrons are the capacitors as parallel plates, the outer skin cover of atoms.
Product details
• ASIN ‏ : ‎ B0CCLPTBDG
• Publication date ‏ : ‎ July 21, 2023
• Language ‏ : ‎ English
• File size ‏ : ‎ 788 KB
• Text-to-Speech ‏ : ‎ Enabled
• Screen Reader ‏ : ‎ Supported
• Enhanced typesetting ‏ : ‎ Enabled
• X-Ray ‏ : ‎ Not Enabled
• Word Wise ‏ : ‎ Enabled
• Sticky notes ‏ : ‎ On Kindle Scribe
• Print length ‏ : ‎ 168 pages

Read my recent posts in peace and quiet. If you, the reader, is wondering why AP posts this to a thread which is off topic in sci.math or sci.physics, is because some stalkers track AP, such as kibo, dan, jan who have been paid to stalk for 3 decades and when they see AP trying to post to his own thread that is on-topic they throw a impossible reCAPTcha suppression and repression at me that only wastes my time. From what AP can make out-- Google is not the only one using reCAPTcha, apparently the US govt rents out reCAPTcha. So if you see a AP post in a thread off topic, is because kooks of reCAPTcha are making it impossible for AP to post to the on-topic thread.

Read all of AP's post in peace and quiet in his newsgroup-- what sci.physics and sci.math should look like when govt spammers are not allowed in a newsgroup to wreck the newsgroup. Govt spammers have their agenda of drag net spam, and then their agenda of spy message codes, such as the "i sick, i cry" baloney, which only ends up ruining the newsgroups and why Google decided to close shop having fought govt bureaucrat mind sets for 30 years, and time to close shop.

AP kindly asks Google to let AP run all three, sci.math, sci.physics, PAU as he runs PAU, now--- all pure science, no spam and no govt b.s.

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Archimedes Plutonium