Ordinals

Does the set of all ordinals exist within ZF?

quasi

2014-04-19 07:59:24 UTC

Post by William Elliot
Does the set of all ordinals exist within ZF?

It's too big to be a set.

quasi

quasi

2014-04-19 08:11:25 UTC

Post by William Elliot
Does the set of all ordinals exist within ZF?

It's too big to be a set.

Hmmm ...

It's certainly not a set in ZFC.

I'm not sure if the "too big" criterion can be applied in ZF.

But how would you _define_ such a set?

Wouldn't the postulated existence of such a set fall victim to
Russell's paradox?

quasi

Peter Percival

2014-04-19 09:51:35 UTC

Post by William Elliot
Does the set of all ordinals exist within ZF?

It's too big to be a set.

Hmmm ...
It's certainly not a set in ZFC.
I'm not sure if the "too big" criterion can be applied in ZF.
But how would you _define_ such a set?
Wouldn't the postulated existence of such a set fall victim to
Russell's paradox?
quasi

Burali-Forti rather than Russell. If there was a set of all ordinals it
would be an ordinal and thus a member of itself, and thus greater than
itself.

--
...if someone seduced my daughter it would be damaging and horrifying
but not fatal. She would recover, marry and have lots of children...
On the other hand, if some elderly, or not so elderly, schoolmaster
seduced one of my sons and taught him to be a homosexual, he would ruin
him for life. That is the fundamental distinction. -- Lord Longford

markus...@gmail.com

2024-02-20 19:35:18 UTC

Post by William Elliot
Does the set of all ordinals exist within ZF?

It's too big to be a set.

Hmmm ...
It's certainly not a set in ZFC.
I'm not sure if the "too big" criterion can be applied in ZF.
But how would you _define_ such a set?
Wouldn't the postulated existence of such a set fall victim to
Russell's paradox?
quasi

No set can contain itself. The set of all ordinals would be a new ordinal, and thus contain itself. Ergo, there cannot be a set of all ordinals.

mitchr...@gmail.com

2024-02-20 19:59:36 UTC

The phrase the "The set of all ordinals" is meaningless if ordinals
are not sets itself. At the time of Burali-Forti set theory was
not that evolved. And the proof at that time didn't use regularity axiom.
So the set of all ordinals was a notion of naive set theory, and not
formulated in modern set theory ordinal terminology, but as a
Una questione sui numeri transfiniti
https://zenodo.org/records/2362091/files/article.pdf

Post by ***@gmail.com
No set can contain itself. The set of all ordinals would be a new ordinal

and thus contain itself. Ergo, there cannot be a set of all ordinals.

Man is making more out of ordinals than is there.
He even did it with the Calculus. Einstein pointed it out.
It was part of what he wrote on his death bed.

Ross Finlayson

2024-02-20 20:15:33 UTC

Post by ***@gmail.com

Post by William Elliot
Does the set of all ordinals exist within ZF?

It's too big to be a set.

Hmmm ...
It's certainly not a set in ZFC.
I'm not sure if the "too big" criterion can be applied in ZF.
But how would you _define_ such a set?
Wouldn't the postulated existence of such a set fall victim to
Russell's paradox?
quasi

No set can contain itself. The set of all ordinals would be a new ordinal, and thus contain itself. Ergo, there cannot be a set of all ordinals.

"... in set-theories like ZF
that are ordinary/well-founded,
according to an axiom like Regularity
of restriction of comprehension."

There are others, ..., "Mengenlehre(n)".

Ross Finlayson

2024-02-20 20:27:06 UTC

Post by ***@gmail.com

Post by William Elliot
Does the set of all ordinals exist within ZF?

It's too big to be a set.

Hmmm ...
It's certainly not a set in ZFC.
I'm not sure if the "too big" criterion can be applied in ZF.
But how would you _define_ such a set?
Wouldn't the postulated existence of such a set fall victim to
Russell's paradox?
quasi

No set can contain itself. The set of all ordinals would be a new
ordinal, and thus contain itself. Ergo, there cannot be a set of all
ordinals.

"... in set-theories like ZF
that are ordinary/well-founded,
according to an axiom like Regularity
of restriction of comprehension."
There are others, ..., "Mengenlehre(n)".

(The set of all ordinals has a name, it's "ORD",
the order type of ordinals, and set of ordinals.)

(One time I wrote a couple different ways to
define, the, "group of all groups", for algebra,
like "GRP".)

(There are wide varieties of, "mothers of all wavelets".)

Mostly these sorts considerations are
called "ZF with Classes" or "ZFC with Classes",
that the Classes or Klassen, if that's right,
are sets, when, you know, they're not sets.

I called it the "Group-Noun Game", because,
it eventually runs out of Group Nouns.

Someone like Quine calls the classes that aren't
sets, "ultimate" classes, while usually the name
for the classes that aren't sets are "proper", classes,
while in some considerations there can only be one,
"proper" class, because, it's as an "absolute", class.

So, after ZFC there's things like NBG, "Neumann-Bernays-Goedel",
or GBN, "Goedel-Bernays-Neumann", who, depending on who you
ask and how formalist they are that day, are or aren't,
ZFC with classes and/or a conservative extension of ZFC.

Jim Burns

2024-02-20 22:02:26 UTC

Post by ***@gmail.com
No set can contain itself.
The set of all ordinals would be a new ordinal,
and thus contain itself.
Ergo, there cannot be a set of all ordinals.

"... in set-theories like ZF
that are ordinary/well-founded,
according to an axiom like Regularity
of restriction of comprehension."
There are others, ..., "Mengenlehre(n)".

However,
whatever sets might be,
ordinals would not be ordinals
if they weren't well.ordered by ∈

In any theory in which ordinals are ordinals,
at least the ordinals have finite.descent,
whatever might be true of other sets.

A proposed set.of.all.ordinals which
held itself would not have finite descent.

Ordinals are well.ordered.
Well.ordered.ness can be re.phrased as
transfinite.induction.ness.
(∀α:(∀β<α:P(β))⇒P(α)) ⟹ ∀γ:P(γ)

FD(γ) == "γ has finite descent"

| Assume each ordinal β < α has finite descent.
| ∀β<α:FD(β)
|
| ⟨ α β δ ε ... ⟩ is a strictly.descending sequence
| α > β
| β has finite descent.
| ⟨ β δ ε ... ⟩ is finite
| ⟨ α β δ ε ... ⟩ is finite
| Generalizing over sequences,
| α has finite descent.

Therefore, generalizing over ordinals,
∀α:(∀β<α:FD(β))⇒FD(α)

By transfinite.induction (by well.order),
∀γ:FD(γ)
Each ordinal has finite descent.

Therefore,
the ordinal(?) holding all(?) ordinals
does not hold itself.

Ross Finlayson

2024-02-21 03:36:36 UTC

Post by ***@gmail.com
No set can contain itself.
The set of all ordinals would be a new ordinal,
and thus contain itself.
Ergo, there cannot be a set of all ordinals.

"... in set-theories like ZF
that are ordinary/well-founded,
according to an axiom like Regularity
of restriction of comprehension."
There are others, ..., "Mengenlehre(n)".

However,
whatever sets might be,
ordinals would not be ordinals
if they weren't well.ordered by ∈
In any theory in which ordinals are ordinals,
at least the ordinals have finite.descent,
whatever might be true of other sets.
A proposed set.of.all.ordinals which
held itself would not have finite descent.
Ordinals are well.ordered.
Well.ordered.ness can be re.phrased as
transfinite.induction.ness.
(∀α:(∀β<α:P(β))⇒P(α)) ⟹ ∀γ:P(γ)
FD(γ) == "γ has finite descent"
| Assume each ordinal β < α has finite descent.
| ∀β<α:FD(β)
|
| ⟨ α β δ ε ... ⟩ is a strictly.descending sequence
| α > β
| β has finite descent.
| ⟨ β δ ε ... ⟩ is finite
| ⟨ α β δ ε ... ⟩ is finite
| Generalizing over sequences,
| α has finite descent.
Therefore, generalizing over ordinals,
∀α:(∀β<α:FD(β))⇒FD(α)
By transfinite.induction (by well.order),
∀γ:FD(γ)
Each ordinal has finite descent.
Therefore,
the ordinal(?) holding all(?) ordinals
does not hold itself.

ORD, the order type of ordinals?
The antinomy of Cesare Burali-Forti?

When you theory has a universe,
it's sort of a singular entity,
it is its own powerset and all, ....

If you stick with bounded theories
and adopt an ultra-finitist formalism,
then you might wonder sometime,
where exactly it is all, at?

It's a usual idea for sorts
of "dualist monism",
since for example
Heraklites or Zen Buddhism,
that the universe really is a thing,
and we are in it,
and that the void really is a thing,
and we are in it,
about the same thing.

Because it's a tautology, ....

It's a sort of brachistology.

ORD: that's its name.

Mild Shock

2024-02-21 13:47:08 UTC

**********************************************************************
Welcome to brain gymnastics about the "class" and "set" distinction.
**********************************************************************

"Ord" is the predication whether a class is transitive and is
well-ordered by the membership relation.

"On" usually denotes the class of sets that are ordinal.
On itself is ordinal, although not set-like.

Its basically the first example of an ordinal in every
set theory, which is not set-like. See also:

See also thes theorems here:

⊢ ¬ On ∈ V
https://us.metamath.org/mpeuni/onprc.html

⊢ Ord On
https://us.metamath.org/mpeuni/ordon.html

And this definition here:

⊢ On = {𝑥 ∣ Ord 𝑥}
https://us.metamath.org/mpeuni/df-on.html

Post by Ross Finlayson
ORD, the order type of ordinals?
The antinomy of Cesare Burali-Forti?

Mild Shock

2024-02-21 13:50:48 UTC

BTW: Ord is prima facie a higher order logic predicate (HOL),
it is not from first order logic (FOL), because it takes a

class argument. But you might rewrite it to FOL for some
kind of arguments sometimes. Its defined here:

⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
https://us.metamath.org/mpeuni/df-ord.html

Post by Mild Shock
**********************************************************************
Welcome to brain gymnastics about the "class" and "set" distinction.
**********************************************************************
"Ord" is the predication whether a class is transitive and is
well-ordered by the membership relation.
"On" usually denotes the class of sets that are ordinal.
On itself is ordinal, although not set-like.
Its basically the first example of an ordinal in every
⊢ ¬ On ∈ V
https://us.metamath.org/mpeuni/onprc.html
⊢ Ord On
https://us.metamath.org/mpeuni/ordon.html
⊢ On = {𝑥 ∣ Ord 𝑥}
https://us.metamath.org/mpeuni/df-on.html

Post by Ross Finlayson
ORD, the order type of ordinals?
The antinomy of Cesare Burali-Forti?

Mild Shock

2024-02-21 14:16:14 UTC

Higher order logic (HOL) was already in use among logicians
when Gödel wrote this booklet:

Kurt Gödel: The Consistency of the Axiom of Choice and of
the Generalized Continuum Hypothesis with the Axioms of
Set Theory, Annals of Mathematical Studies, Volume 3, Princeton NJ, 1940
https://www.amazon.com/dp/0691079277

Meta math is not so open about it that it uses HOL.
Using Neumann-Bernays-Gödel-Mengenlehre (NBG)
might also not help much. Pocking into Isabelle/HOL wasn't

so satisfactory either, they often work with α set type
constructor, so that the set theory and all theorems have
a type parameter α. But Meta math looks very cute,

is less a eyesore than anything else.

Post by Mild Shock
BTW: Ord is prima facie a higher order logic predicate (HOL),
it is not from first order logic (FOL), because it takes a
class argument. But you might rewrite it to FOL for some
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
https://us.metamath.org/mpeuni/df-ord.html

Post by Ross Finlayson
ORD, the order type of ordinals?
The antinomy of Cesare Burali-Forti?

Mild Shock

2024-02-22 08:03:59 UTC

Seriously, you don't know what classes are?

The membership relation is the same
for members of classes and for members of sets.
Since members of classes are sets just like

the members of sets are sets, in ZF. And there is
only one membership relation ∈ between sets. The
distinction between classes and sets was described

in the past as:

sets: includes collections of sizes from the numbers to
the transfinite numbers
classes: includes collections that Cantor called
NCONSISTENT MULTIPLICITIES

Post by Ross Finlayson
Of course, the goal is "there are no paradoxes at all",
then what seem "inconsistent multiplicities", just don't relate.
If ORD involves class/set distinction,
and a set-theory can also be written as a part-theory,
then what's part/particle distinction/
If set theory's relation is "elt", element-of, "in"
and class theory's relation is "members", "contains", "has",
class/set theory
set/part theory?
Here that "numbering" and "counting" are two different things,
one for ordering theory the other for collection,
ordinals and sets, numbering and counting,
what about
set/class distinction and
set/part distinction and
part/class distinction?
See, this is among reasons why
I've been way both ahead of
and on top of this for a long time,
and trying to tell you so all the time.
I told you, ..., I told you.
Mostly is for understanding that
"numbering" and "counting" are
two different things, and they
involve each other in their resources.

Mild Shock

2024-02-22 08:11:29 UTC

In the philosophy of mathematics, specifically the philosophical
foundations of set theory, limitation of size is a concept developed by
Philip Jourdain and/or Georg Cantor to avoid Cantor's paradox. It
identifies certain "inconsistent multiplicities", in Cantor's
terminology, that cannot be sets because they are "too large". In modern
terminology these are called proper classes.
https://en.wikipedia.org/wiki/Limitation_of_size

You might like this book:

Cantor's ideas formed the basis for set theory and also for the
mathematical treatment of the concept of infinity. The philosophical and
heuristic framework he developed had a lasting effect on modern
mathematics, and is the recurrent theme of this volume. Hallett explores
Cantor's ideas and, in particular, their ramifications for
Zermelo-Frankel set theory.
https://academic.oup.com/pq/article-abstract/36/144/429/1567519

Post by Mild Shock
Seriously, you don't know what classes are?
The membership relation is the same
for members of classes and for members of sets.
Since members of classes are sets just like
the members of sets are sets, in ZF. And there is
only one membership relation ∈ between sets. The
distinction between classes and sets was described
sets: includes collections of sizes from the numbers to
the transfinite numbers
classes: includes collections that Cantor called
NCONSISTENT MULTIPLICITIES

Ross Finlayson

2024-02-22 18:16:59 UTC

Actually, for class/set distinction,
I just introduced set/part distinction,
and part/particle distinction,
and set/particle distinction.

set:class::part:particle

set:part::class:particle

This is a usual form that A:B::C:D is
that A relates to B as C relates to D,
"set is to class as part is to particle", and
"set is to part as class is to particle".

Mild Shock

2024-02-22 18:20:16 UTC

Doesn't make any sense at all.

Not a single mention of proper classes here:
https://plato.stanford.edu/ENTRIES/mereology/

Post by Ross Finlayson
Actually, for class/set distinction,
I just introduced set/part distinction,
and part/particle distinction,
and set/particle distinction.
set:class::part:particle
set:part::class:particle
This is a usual form that A:B::C:D is
that A relates to B as C relates to D,
"set is to class as part is to particle", and
"set is to part as class is to particle".

Mild Shock

2024-02-22 18:22:45 UTC

I guess we have reached your intellectual
boundaries, inherent in your squirell brain
sized, that of a walnut, cerebrum and cerebrellum.

Post by Mild Shock
Doesn't make any sense at all.
https://plato.stanford.edu/ENTRIES/mereology/

Mild Shock

2024-02-22 18:40:27 UTC

What we can say is the following:

i) Every set is a class
ii) Not every class is a set

So there is a hypernym / hyponym relationship
between the two. Here is are proof of i) and ii):

Proof i): Let s be a set. Then we can form
the class { x | x e s }. So there is an injection
from the sets to the classes.

Proof ii): Let V be the class { x | true },
this is the universal class which is provably
not a set. So there is no surjection from
the sets to the classes.

Hope this helps. Injection is usually taken
as indicative that two sets are in the
less than or equal relation ship, i.e. ⊆.
And lack of surjection indicates that there
is no bijection, i.e. ≠, so we have:

Sets ⊆ Classes and Sets ≠ Classes

Or together:

Sets ⊂ Classes

The difference Class \ Sets, those things
that are classes but not sets, are called
proper classes.

Post by Mild Shock
I guess we have reached your intellectual
boundaries, inherent in your squirell brain
sized, that of a walnut, cerebrum and cerebrellum.

Post by Mild Shock
Doesn't make any sense at all.
https://plato.stanford.edu/ENTRIES/mereology/

Mild Shock

2024-02-22 19:00:33 UTC

Better symbolism would be:

Sets' ⫋ Classes

Where Sets' results from Sets by the injection
{ x | x e s } for each x e Sets. This gives a little
transfer principle. If you can prove, i.e. that

a property holds for all classes:

∀X P(X)

Then it follows, that the property holds for all sets.

∀x P(x)

Proof: In higher order logic one would probably
write λy.(x y) for { y | y e x }, by eta reduction
we have λy.(x y)= x, so one can prove:

∀X P(X)
------------ (∀ elim)
P(λy.(x y))
------------ (η-reduction)
P(x)
------------ (∀ Intro)
∀x P(x)

Q.E.D.

η-reduction expresses the idea of extensionality
https://en.wikipedia.org/wiki/Lambda_calculus#%CE%B7-reduction

Wao! Now I did a lot of cheating, sweeping a lot of
details under the rug. I guess this is not
the standard way to do these things.

Better have a look here:

Basic Set Theory - Azriel Levy
https://www.amazon.com/dp/0486420795

Post by Mild Shock
i) Every set is a class
ii) Not every class is a set
So there is a hypernym / hyponym relationship
Proof i): Let s be a set. Then we can form
the class { x | x e s }. So there is an injection
from the sets to the classes.
Proof ii): Let V be the class { x | true },
this is the universal class which is provably
not a set. So there is no surjection from
the sets to the classes.
Hope this helps. Injection is usually taken
as indicative that two sets are in the
less than or equal relation ship, i.e. ⊆.
And lack of surjection indicates that there
Sets ⊆ Classes and Sets ≠ Classes
Sets ⊂ Classes
The difference Class \ Sets, those things
that are classes but not sets, are called
proper classes.

Post by Mild Shock
I guess we have reached your intellectual
boundaries, inherent in your squirell brain
sized, that of a walnut, cerebrum and cerebrellum.

Post by Mild Shock
Doesn't make any sense at all.
https://plato.stanford.edu/ENTRIES/mereology/

Mild Shock

2024-02-22 19:08:34 UTC

Are you pulling a John Gabriel? Back
to greek ratios. Euclids general form:

A : B = C : D

What about this form:

A : B : C = E : F : G
http://aleph0.clarku.edu/~djoyce/elements/bookV/defV3.html

I would especially recommend equations of the form:

cornet:walnut:pistachio
= cup:banana:mango
= hotday:ingest:cooling

Hm. "squirrel:brain::walnut:cerebrum".

Ross Finlayson

2024-02-22 19:01:00 UTC

Post by Mild Shock
Doesn't make any sense at all.
plato.stanford.edu/ENTRIES/mereology/

Hm. "squirrel:brain::walnut:cerebrum".

(The spell-checker there seems omitted 'cerebellum',
and didn't read into the theory of parts some
'ultimate particles', perhaps it might help to
addend an entry 'atomism'. Here though it's mathematics
and we have this entire canonical exposition about
"uniform and continuous time", "Zeno's arguments".
Maybe that will add some more context for the
"brain of a squirrel" simile, metaphor.)

With regards to intensionality and extensionality,

intensionality <- structure-equals
extensionality <- duck-type-equals

simile <- declaring-relates
metaphor <- not-equals-so-relate-to-relates

and simile and metaphor, and a strong theory of types,
one can make a simile and metaphor out of pretty much
anything, that of course there's a usual idea that
"relevance logic" dictates relevance for a strong
theory of types.

My, what a rude person. It's kind of like in the old days,
when you'd see something doing something or not demonstrating
knowledge of something, one might say, "what are you doing,
you big dummy". This was like, "hey, you big dummy, what do
you think you are doing". Or, like to a dog, "aw, you big dummy".

Here I don't quite get that kind of attitude in a
conversation about the fundamental objects of the
universe of the fundamental objects of mathematics,
about the theory of the objects of the fundamental
theory of the fundamental objects of mathematics.

It's like, "you're going to have to be a bigger dummy".

I.e., the universe of mathematical objects
"is what it is", it's got numbers in it, and on it,
and counting is an act, two different things.

Mild Shock

2024-02-22 19:11:40 UTC

If you follow this thinking long enough,
you will proof Mückenheims identifiable numbers.
And suddently have finite ascent in mathematics.

LoL

Post by Ross Finlayson
I.e., the universe of mathematical objects
"is what it is", it's got numbers in it, and on it,
and counting is an act, two different things.

Mild Shock

2024-02-22 19:13:38 UTC

Wolfgang Mückenheim copy cats are rather boring.

Post by Mild Shock
If you follow this thinking long enough,
you will proof Mückenheims identifiable numbers.
And suddently have finite ascent in mathematics.
LoL

Post by Ross Finlayson
I.e., the universe of mathematical objects
"is what it is", it's got numbers in it, and on it,
and counting is an act, two different things.

Jim Burns

2024-02-22 19:34:56 UTC

Post by Mild Shock

[...]

Seriously, you don't know what classes are?
The membership relation is the same
for members of classes and for members of sets.
Since members of classes are sets just like
the members of sets are sets, in ZF. And there is
only one membership relation ∈ between sets. The
distinction between classes and sets was described
sets: includes collections of sizes from the numbers to
the transfinite numbers
classes: includes collections that Cantor called
NCONSISTENT MULTIPLICITIES

I once had a lecturer in math who would refer to
proper classes as "syntactic sugar",
which I found out today is a _programming_ term.

https://en.wikipedia.org/wiki/Syntactic_sugar
| In computer science, syntactic sugar is syntax within
| a programming language that is designed to make things
| easier to read or to express. It makes the language
| "sweeter" for human use: things can be expressed
| more clearly, more concisely, or in an alternative style
| that some may prefer.

I take that to mean that
| α ∈ ORD
is a "sweeter" way to say something like
| α is a regular transitive set of transitive sets
| Reg(α) ∧ Trans(α) ∧ ∀β ∈ α: Trans(β)

Reg(α) ⟺
α ≠ ∅ ⟹ ∃β ∈ α: β∩α = ∅

Trans(α) ⟺
∀β,γ: γ ∈ β ∧ β ∈ α ⟹ γ ∈ α

On the one hand,
questions surrounding the existence of ORD
have a lower bar to clear,
but
some things which a set can do
proper class ORD can't do.

Or that is my impression.
Caveat lector.

Ross Finlayson

2024-02-22 19:39:33 UTC

Post by Mild Shock

[...]

Seriously, you don't know what classes are?
The membership relation is the same
for members of classes and for members of sets.
Since members of classes are sets just like
the members of sets are sets, in ZF. And there is
only one membership relation ∈ between sets. The
distinction between classes and sets was described
sets: includes collections of sizes from the numbers to
the transfinite numbers
classes: includes collections that Cantor called
NCONSISTENT MULTIPLICITIES

I once had a lecturer in math who would refer to
proper classes as "syntactic sugar",
which I found out today is a _programming_ term.
https en.wikipedia.org/wiki/Syntactic_sugar
| In computer science, syntactic sugar is syntax within
| a programming language that is designed to make things
| easier to read or to express. It makes the language
| "sweeter" for human use: things can be expressed
| more clearly, more concisely, or in an alternative style
| that some may prefer.
I take that to mean that
| α ∈ ORD
is a "sweeter" way to say something like
| α is a regular transitive set of transitive sets
| Reg(α) ∧ Trans(α) ∧ ∀β ∈ α: Trans(β)
Reg(α) ⟺
α ≠ ∅ ⟹ ∃β ∈ α: β∩α = ∅
Trans(α) ⟺
∀β,γ: γ ∈ β ∧ β ∈ α ⟹ γ ∈ α
On the one hand,
questions surrounding the existence of ORD
have a lower bar to clear,
but
some things which a set can do
proper class ORD can't do.
Or that is my impression.
Caveat lector.

If it helps,
which is always usually an implicit,
"if it helps", "take it or leave it",
"there's food", "caveat lector",
if it helps, when I learned about
class/set distinction and started
to explore its concepts, it's been
since about 20-25 years, and you
can largely read my responsive developments
about it on archives of sci.math and sci.logic.

So, class/set distinction, and _proper_ classes,
gets into things like "there can be only one",
that all the proper classes get conflated into one.

So, the term "inconsistent multiplicity",
here has been in the context since Y2K, say.

Now, for ORD, or even just N,
that N = N+1 or N = N+,
it's just about the most usual thing
since you've ever known the word "infinity".

(For "great minds think alike" and for
somehow "the greatest mind, thinks, the same",
I'm impressed that our synchronicity arrived
at "caveat lector", independently. Also,
"know the food" and "thoroughly chew the food".)

In the "naive", set theory, what aren't "ordinary",
sets, are proper classes, or as Quine puts it,
ultimate classes, as with respect to there being
one at all, some "absolute class".

Mild Shock

2024-02-22 23:41:20 UTC

This is the school that tries to avoid
higher order logic (HOL). And then sneaks in
classes as a kind of syntactic sugar into

FOL, with this rule:

x e { y | p(y) } <=> p(x)

If you are not wearing this chastity belt, its
just some HOL juggling. Like here:

Ross Finlayson schrieb:
https://leanprover-community.github.io/mathlib_docs/set_theory/zfc/basic.html#Class

But I suggest to study simple types first. And
then maybe dependent types. To understand the
type theoretic capture of set theory.

See also:

Should Type Theory Replace Set Theory as
the Foundation of Mathematics?

13 February 2023 - Thorsten Altenkirch
https://link.springer.com/article/10.1007/s10516-023-09676-0

Post by Jim Burns
I once had a lecturer in math who would refer to
proper classes as "syntactic sugar",
which I found out today is a _programming_ term.

Jim Burns

2024-02-21 17:59:50 UTC

Post by Jim Burns
Ordinals are well.ordered.

Only those which can be specified.

No.
All of them are well.ordered.
Anything else wouldn't be the ordinals.

Anything else would be like declaring
that only specifiableᵂᴹ right.triangles
have three corners.

Post by Jim Burns
In any theory in which ordinals are ordinals,
at least the ordinals have finite.descent,

That proves finite ascend too,
because
otherwise
every ordinal could be ascended
and then
the way upstairs could be gone back downstairs.

The ordinals' descents and ascents are not the same.

If any of an ordinal's descents is infinite,
the ordinal doesn't have finite.descent.

If any of an ordinal's ascents is infinite,
the ordinal doesn't have finite.ascent.

Each ordinal α has a successor α+1
α+1 has α+2, etc.

For ordinal a
⟨ α α+1 α+2 α+3 ... ⟩ is an infinite ascent.
α doesn't have finite.ascent.

Generalizing over ordinals,
no ordinal a has finite.ascent.

For each ordinal ψ
if ψ has any infinite descent,
then, because well.order,
an ordinal χ exists first with any infinite descent.

However,
one step down from χ to any ordinal β < χ is to
β with only finite descents,
and finite plus one is finite.
First χ doesn't have any infinite descent.
Contradiction.

ψ doesn't have an infinite descent.

Generalizing over ordinals,
each ordinal ψ has finite.descent.

Jim Burns

2024-02-22 16:13:26 UTC

Post by Jim Burns
Ordinals are well.ordered.

Only those which can be specified.

No.
All of them are well.ordered.

How do you know?

In the same way that I know
that right.triangles have three corners.
Not by looking at ordinals.
By knowing what "ordinal" means.

Post by Jim Burns
Anything else wouldn't be the ordinals.

In fact, not these ordinals.

Then "these ordinals" are like
four.cornered right.triangles.

Post by Jim Burns
Anything else would be like declaring
that only specifiableᵂᴹ right.triangles
have three corners.

That is too drastic.
Natnumbers keep almost all of their properties.

Four.cornered right.triangles share
many properties with other plane figures.
They aren't right.triangles.

Post by Jim Burns
The ordinals' descents and ascents are not the same.

Every way up can be reversed.

An infinite way up isn't _to_ any ordinal.
Reversed, it isn't _from_ any ordinal.

That proves that also the ascents are finite.

By excluding all infinite ascents,
we can prove that all _remaining_ ascents
are finite.

...which doesn't deny that
⟨ α α+1 α+2 α+3 ... ⟩ is an infinite ascent.

In contrast,
there is no first ordinal with
an infinite descent, so
there is no ordinal with
an infinite descent.

Post by Jim Burns
For each ordinal ψ
if ψ has any infinite descent,
then, because well.order,
an ordinal χ exists first with any infinite descent.
However,
one step down from χ to any ordinal β < χ is to
β with only finite descents,
and finite plus one is finite.
First χ doesn't have any infinite descent.
Contradiction.
ψ doesn't have an infinite descent.

And one step upwards is finite too.
Finite plus one is finite.a

Finite ascents exist.
Finite descents exist too.

Infinite ascents exist,
⟨ α α+1 α+2 α+3 ... ⟩
A first infinite descent is
a contradiction.

ψ doesn't have an infinite ascent
(for every visible predecessor).

Daek numbers wouldn't make
⟨ ψ ψ+1 ψ+2 ψ+3 ... ⟩ less infinite.

Post by Jim Burns
Generalizing over ordinals,
each ordinal ψ has finite.descent.

Each ordinal has finite ascent.

Know what "finite ascent" means:
no infinite ascents.

2024-02-23 08:47:07 UTC

Post by Jim Burns
Ordinals are well.ordered.

Only those which can be specified.

No.
All of them are well.ordered.

How do you know?

In the same way that I know
that right.triangles have three corners.

Yes, you are right. I exaggerated. Having three corners is essential for
triangles. Being well-ordered is essential for ordinals. What I meant is
that we cannot follow the well-order into the dark realm. In particular
Peano ceases.

Regards, WM

Mild Shock

2024-02-26 11:58:52 UTC

I always thought all platonic realms are flooded with light!

Now something from the formalists that want to
to capture the light. It seems to me ordinals

are a forgotten treasure. Nice papers here:

W. A. Howard: Assignment of ordinals to terms for
primitive recursive functionals of finite type.
Howard, William A. (September 1970)
https://www.sciencedirect.com/science/article/abs/pii/S0049237X08707705

Systems of logic based on ordinals
Alan Touring - 1938 (sic!)
https://people.math.ethz.ch/~halorenz/4students/Literatur/TuringOrdinalLogic.pdf

Homework for Dan Christensen:

- Show PA consistent.

Post by Jim Burns
Ordinals are well.ordered.

Only those which can be specified.

No.
All of them are well.ordered.

How do you know?

In the same way that I know
that right.triangles have three corners.

Jim Burns

2024-02-27 19:25:42 UTC

...]

Having three corners is essential for triangles.
Being well-ordered is essential for ordinals.
What I meant is that
we cannot follow the well-order into the dark realm.
In particular Peano ceases.

"Following the well.order" is a metaphor.

I am not a Form..
I've never met you. Still, I'm confident that
you aren't a Form, either.
We aren't in the same Realm as the ordinals.
Un.metaphorically, in our Realm,
we have no opportunity to follow or to not.follow
the well.order.

What I gather that you (WM) are asserting by metaphor
is a wrong assertion.

Having three corners is essential for triangles.
Being well-ordered is essential for ordinals.
What I meant is that
we cannot follow the well-order into the dark realm.
In particular Peano ceases.

We know that Peano induction not.ceases in
the Peano (final) ordinals.

It is a mistake to expect more than
Peano induction not.ceasing in
the Peano (final) ordinals.

We know that transfinite induction not.ceases in
the transfinite ordinals.

It is a mistake to expect more than
transfinite induction not.ceasing in
the transfinite ordinals.

We know that
being well.ordered not.ceases in
the transfinite ordinals.
Even beyond the first _inaccessible_ ordinal κ
being well.ordered not.ceases.

https://en.wikipedia.org/wiki/Inaccessible_cardinal
| In set theory, an uncountable cardinal is inaccessible
| if it cannot be obtained from smaller cardinals by
| the usual operations of cardinal arithmetic.

We know that because
transfinite induction is well.order in drag.
Thus, just like well.order,
transfinite induction not.ceases in the ordinals.

----
Well.ordering not.ceases in the ordinals.

If exists ordinal γ: p(γ)
then exists first ordinal β: p(β)

∃ᵒʳᵈγ:p(γ) ⟹ ∃#1ᵒʳᵈβ:p(β)

∃ᵒʳᵈγ:p(γ) ⟹ ∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α))

¬∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α)) ⟹ ¬∃ᵒʳᵈγ:p(γ)

∀ᵒʳᵈβ:(¬p(β) ∨ ¬∀ᵒʳᵈα<β:¬p(α)) ⟹ ∀ᵒʳᵈγ:¬p(γ)

∀ᵒʳᵈβ:(̅p(β) ∨ ¬∀ᵒʳᵈα<β:̅p(α)) ⟹ ∀ᵒʳᵈγ:̅p(γ)

∀ᵒʳᵈβ:(∀ᵒʳᵈα<β:̅p(α) ⇒ ̅p(β)) ⟹ ∀ᵒʳᵈγ:̅p(γ)

∀ᵒʳᵈβ:(̅pᣔ[0,β) ⇒ ̅pᣔ[0,β⁺¹)) ⟹ ∀ᵒʳᵈγ:̅p(γ)

where
̅pᣔ[0,β) ⟺ ∀ᵒʳᵈα<β:̅p(α)
̅pᣔ[0,β) ∧ ̅p(β) ⟺ ̅pᣔ[0,β⁺¹)

If, for each ordinal β
̅p.before.β implies ̅p.before.β⁺¹
then, for each ordinal γ ̅p

Transfinite induction not.ceases in the ordinals.

Ross Finlayson

2024-02-27 19:59:39 UTC

...]

Having three corners is essential for triangles.
Being well-ordered is essential for ordinals.
What I meant is that
we cannot follow the well-order into the dark realm.
In particular Peano ceases.

"Following the well.order" is a metaphor.
I am not a Form..
I've never met you. Still, I'm confident that
you aren't a Form, either.
We aren't in the same Realm as the ordinals.
Un.metaphorically, in our Realm,
we have no opportunity to follow or to not.follow
the well.order.
What I gather that you (WM) are asserting by metaphor
is a wrong assertion.

Having three corners is essential for triangles.
Being well-ordered is essential for ordinals.
What I meant is that
we cannot follow the well-order into the dark realm.
In particular Peano ceases.

We know that Peano induction not.ceases in
the Peano (final) ordinals.
It is a mistake to expect more than
Peano induction not.ceasing in
the Peano (final) ordinals.
We know that transfinite induction not.ceases in
the transfinite ordinals.
It is a mistake to expect more than
transfinite induction not.ceasing in
the transfinite ordinals.
We know that
being well.ordered not.ceases in
the transfinite ordinals.
Even beyond the first _inaccessible_ ordinal κ
being well.ordered not.ceases.
https://en.wikipedia.org/wiki/Inaccessible_cardinal
| In set theory, an uncountable cardinal is inaccessible
| if it cannot be obtained from smaller cardinals by
| the usual operations of cardinal arithmetic.
We know that because
transfinite induction is well.order in drag.
Thus, just like well.order,
transfinite induction not.ceases in the ordinals.
----
Well.ordering not.ceases in the ordinals.
If exists ordinal γ: p(γ)
then exists first ordinal β: p(β)
∃ᵒʳᵈγ:p(γ) ⟹ ∃#1ᵒʳᵈβ:p(β)
∃ᵒʳᵈγ:p(γ) ⟹ ∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α))
¬∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α)) ⟹ ¬∃ᵒʳᵈγ:p(γ)
∀ᵒʳᵈβ:(¬p(β) ∨ ¬∀ᵒʳᵈα<β:¬p(α)) ⟹ ∀ᵒʳᵈγ:¬p(γ)
∀ᵒʳᵈβ:(̅p(β) ∨ ¬∀ᵒʳᵈα<β:̅p(α)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
∀ᵒʳᵈβ:(∀ᵒʳᵈα<β:̅p(α) ⇒ ̅p(β)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
∀ᵒʳᵈβ:(̅pᣔ[0,β) ⇒ ̅pᣔ[0,β⁺¹)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
where
̅pᣔ[0,β) ⟺ ∀ᵒʳᵈα<β:̅p(α)
̅pᣔ[0,β) ∧ ̅p(β) ⟺ ̅pᣔ[0,β⁺¹)
If, for each ordinal β
̅p.before.β implies ̅p.before.β⁺¹
then, for each ordinal γ ̅p
Transfinite induction not.ceases in the ordinals.

Is it not.ultimately.untrue?

That, "the limit is the sum" has that,
the sum is no different than the limit,
and, the expression is only the sum in the limit,
and, not in the (infinite) limit,
the expression is not necessarily the sum.

Zeno's still there on the bridge,
and every time you approach,
he walks right up to the entrance to the bridge,
and demands, "how do you get _all the way across_"?

And you say "I can get almost there, ...".

And he says, "that's funny, I just came from there,
and you weren't there".

2024-02-27 20:05:00 UTC

Post by Jim Burns
We know that Peano induction not.ceases in
the Peano (final) ordinals.

We know that every visible step is reversible.

Regardes, WM

Jim Burns

2024-02-27 22:24:32 UTC

Post by Jim Burns
We know that Peano induction not.ceases in
the Peano (final) ordinals.

We know that every visible step is reversible.

A one.ended ascent is reversible,
but is not a descent.

A one.ended ascent starts and not.stops.

A reverse one.ended ascent stops and not.starts.

A descent starts.

A reverse one.ended ascent isn't a descent.

----
No ordinal is first to start a one.ended descent.

No ordinal starts a one.ended descent.

----
ℕ̲⇊ is the set of final ordinals.

For each ordinal ξ
{ξ+n: n ∈ ℕ̲⇊} is
a darkᵂᴹfree one.ended ascent from ξ

2024-02-28 09:48:38 UTC

Post by Jim Burns
We know that Peano induction not.ceases in
the Peano (final) ordinals.

We know that every visible step is reversible.

A one.ended ascent is reversible,
but is not a descent.

Every reversion of an ascent is a descent.

Regards, WM

Jim Burns

2024-02-28 11:52:27 UTC

Post by Jim Burns
We know that Peano induction not.ceases in
the Peano (final) ordinals.

We know that every visible step is reversible.

A one.ended ascent is reversible,
but is not a descent.

Every reversion of an ascent is a descent.

A one.ended ascent starts and not.stops.

A reverse one.ended ascent stops and not.starts.

A descent starts.

A reverse one.ended ascent isn't a descent.

2024-02-28 17:24:48 UTC

Post by Jim Burns
We know that Peano induction not.ceases in
the Peano (final) ordinals.

We know that every visible step is reversible.

A one.ended ascent is reversible,
but is not a descent.

Every reversion of an ascent is a descent.

A one.ended ascent starts and not.stops.

As long as it runs through visible numbers it is finite and reversible.

Regards, WM

Richard Damon

2024-02-28 22:07:55 UTC

Post by Jim Burns
We know that Peano induction not.ceases in
the Peano (final) ordinals.

We know that every visible step is reversible.

A one.ended ascent is reversible,
but is not a descent.

Every reversion of an ascent is a descent.

A one.ended ascent starts and not.stops.

As long as it runs through visible numbers it is finite and reversible.
Regards, WM

In other words, your definition of "Visible Numbers" are finite sub-sets
of the actual set of values.

And this is because your logic can only handle finite sets.

Your "Dark" numbers are just the numbers that you can not handle with
your restricted finite limited logic.

There is actually no problem with those numbers, except you incorrectly
limited logic can't deal with them.

They are a product of YOUR limitations, not of the inability of the
proper logic to deal with them.

2024-02-29 08:23:09 UTC

Post by Jim Burns
We know that Peano induction not.ceases in
the Peano (final) ordinals.

We know that every visible step is reversible.

A one.ended ascent is reversible,
but is not a descent.

Every reversion of an ascent is a descent.

A one.ended ascent starts and not.stops.

As long as it runs through visible numbers it is finite and reversible.

In other words, your definition of "Visible Numbers" are finite sub-sets
of the actual set of values.

Visible narural numbers are FISONs {1, 2, 3, ..., n}.

Post by Richard Damon
And this is because your logic can only handle finite sets.

This because there is no infinite natural number.

Post by Richard Damon
Your "Dark" numbers are

the only possibility to have completed infinity. Note: "Going on and on"
is not completed infinity but potential infinity.

Post by Richard Damon
There is actually no problem with those numbers,

What numbers? Infinite natural numbers?

Regards, WM

Richard Damon

2024-02-29 12:35:22 UTC

Post by Jim Burns
We know that Peano induction not.ceases in
the Peano (final) ordinals.

We know that every visible step is reversible.

A one.ended ascent is reversible,
but is not a descent.

Every reversion of an ascent is a descent.

A one.ended ascent starts and not.stops.

As long as it runs through visible numbers it is finite and reversible.

In other words, your definition of "Visible Numbers" are finite
sub-sets of the actual set of values.

Visible narural numbers are FISONs {1, 2, 3, ..., n}.

And so SUBSETS of the Natural Number

Post by Richard Damon
And this is because your logic can only handle finite sets.

This because there is no infinite natural number.

No, but there are an infinte number of them.

You logic can't handle sets of infinite/unbounded size.

Post by Richard Damon
Your "Dark" numbers are

the only possibility to have completed infinity. Note: "Going on and on"
is not completed infinity but potential infinity.

Only because your logic can't handle it.

Post by Richard Damon
There is actually no problem with those numbers,

What numbers? Infinite natural numbers?

The unbounded set of Natural Numbers that go on and on and on.

Post by WM
Regards, WM

You are just proving your logic is broken and your brain is two sizes
too small for what you are trying to do.

2024-02-29 19:25:40 UTC

Post by WM
Visible narural numbers are FISONs {1, 2, 3, ..., n}.

And so SUBSETS of the Natural Number

Of course. What else? Every natural number belongs to the set ℕ.

Post by Richard Damon
There is actually no problem with those numbers,

What numbers? Infinite natural numbers?

The unbounded set of Natural Numbers that go on and on and on.

Visible numbers do never complete infinity.

Regards, WM

Chris M. Thomasson

2024-02-29 21:48:01 UTC

Post by WM
Visible narural numbers are FISONs {1, 2, 3, ..., n}.

And so SUBSETS of the Natural Number

Of course. What else? Every natural number belongs to the set ℕ.

Post by Richard Damon
There is actually no problem with those numbers,

What numbers? Infinite natural numbers?

The unbounded set of Natural Numbers that go on and on and on.

Visible numbers do never complete infinity.

Huh? Visible? Oh god... You are a 100% hyper hardcore ultra finite odd
ball... Infinity makes your brain bleed?

Richard Damon

2024-03-01 03:12:35 UTC

Post by WM
Visible narural numbers are FISONs {1, 2, 3, ..., n}.

And so SUBSETS of the Natural Number

Of course. What else? Every natural number belongs to the set ℕ.

And evvery natural number is finite and thus namable and thus visible.

Post by Richard Damon
There is actually no problem with those numbers,

What numbers? Infinite natural numbers?

The unbounded set of Natural Numbers that go on and on and on.

Visible numbers do never complete infinity.

Right, but no individual Natural Number does either, so they all can be
visible.

It is the SET that "completes infinity", and the set isn't any of the
individual number.

Post by WM
Regards, WM

Ross Finlayson

2024-03-01 03:36:34 UTC

Post by WM
Visible narural numbers are FISONs {1, 2, 3, ..., n}.

And so SUBSETS of the Natural Number

Of course. What else? Every natural number belongs to the set ℕ.

And evvery natural number is finite and thus namable and thus visible.

Post by Richard Damon
There is actually no problem with those numbers,

What numbers? Infinite natural numbers?

The unbounded set of Natural Numbers that go on and on and on.

Visible numbers do never complete infinity.

Right, but no individual Natural Number does either, so they all can be
visible.
It is the SET that "completes infinity", and the set isn't any of the
individual number.

Post by WM
Regards, WM

Why do you keep repeating this if everybody here knows that,
and it's always the case the it just tit-for-tat = for-i-a.
For-git-a-bout-tat, instead of get-tit-for-tat-bout.

A usual idea about the inductive set is that it includes
itself, and yes I know that Russell says it doesn't after
he proves that it does.

The continuum limit of f(n) = n/d in the continuum limit
as d goes to infinity, has that ran(f) has extent, density,
completeness, measure, implies it's a continuous domain,
implies f isn't a Cartesian function, sort of results a
special result between discrete and continuous,
kind of like sliced bread, ....

It's a most usual notion of ordering theory, which is
the theory where ordinals are primary elements instead
of sets being primary elements, to support the time-ordering-property
for the line integral and path integral and the [0,1] of
category theory, and so on, Jordan content, "iota-values".

That is to say, besides you know a usual Riemann, Lebesgue,
Stieltjes, there's Jordan, in measure theory, it's these, ....

Ordinals are primary elements in ordering theory,
sets then implemented in those instead of those in sets,
put them together is sort of how it is, "ubiquitous ordinals",
order type being powerset being successor, this kind of thing.

I imagine and hope you knew that ordering theory has that
ordinals are primary objects in the theory and it's different
than set theory where sets are primary objects in the theory,
and that they're usually enough used to model each other
because they're both so ubiquitous.

Then with regards ".999, read it again", it's like,
you're welcome to read it either way, it being both,
as long as you keep it sorted and straight both ways.

Mild Shock

2024-03-01 19:38:19 UTC

Somehow I am quite new to the ordinal analysis
that Alan Turing started. Nice find Gödel introduces
a nice notion in his incompletness paper,

namely he says a function is primitive recursive,
to degree N, if the primitive recursion schema is
applied N times. So I guess this definition of

factorial would then have degree 3, since 'R' is used 3 times:

/* The R combinator */
natrec(_, 0, X, X) :- !. % attention, not steadfast
natrec(F, N, X, Z) :- M is N-1, natrec(F, M, X, Y), call(F, Y, Z).

plus(X, Y, Z) :- natrec(succ, X, Y, Z).

mult(X, Y, Z) :- natrec(plus(X), Y, 0, Z).

step((X,Y),(Z,T)) :- succ(X,Z), mult(Z,Y,T).

factorial(X,Y) :- natrec(step,X,(0,1),(_,Y)).

Works fine, although eats quite some computing resources,
i.e. 9_303_219 inferences, since it must form
3268800 successors:

?- factorial(10,X).
X = 3628800.

?- time(factorial(10,X)).
% 9,303,219 inferences, 0.578 CPU in 0.617 seconds
(94% CPU, 16092054 Lips)
X = 3628800.

... fourier my ass, what has it to do with ordinals ...

Mild Shock

2024-03-01 19:52:28 UTC

Know that one is the secret and source of all the cardinals.
-- Abraham ibn Ezra (1153)

But have mercy to me. So far I thought in ordinal
anlysis of programs, we would simply take the
tree of execution, and this is somehow an ordinal

for a terminating program? But whats the tree
repectively ordinal for for example 3! = 6 ?
Are all finite ordinals the same or not?

For example the ordinals 0, 1, 2, 3, with von
Neumann succeessor are:

0 = {}
1 = {{}}
2 = {{},{{}}}
3 = {{},{{}},{{{},{{}}}}}.

Or as trees, * = empty set, o = non-empty set:

0 = *

1 = o
|
*

2 = o
/ \
* o
|
*

3 = o
/ | \
* o o
| / \
* * o
|
*

But what about this tree, it has also no infinite decend,
but what property is missing to make it an ordinal?

? = o
/ | \
* o o
| |
* o
|
*

Or as a set:

? = {{},{{}},{{{}}}}.

Why is it not an ordinal?

P.S.: I tried to find an answer here, but I guess
I am too lazy to read it. Its starts with the funny
quote and has funny pictures in it:

Trees, ordinals and termination
N Dershowitz · 1993
https://link.springer.com/content/pdf/10.1007/3-540-56610-4_68.pdf

Post by Mild Shock
Somehow I am quite new to the ordinal analysis
that Alan Turing started. Nice find Gödel introduces
a nice notion in his incompletness paper,
namely he says a function is primitive recursive,
to degree N, if the primitive recursion schema is
applied N times. So I guess this definition of
/* The R combinator */
natrec(_, 0, X, X) :- !. % attention, not steadfast
natrec(F, N, X, Z) :- M is N-1, natrec(F, M, X, Y), call(F, Y, Z).
plus(X, Y, Z) :- natrec(succ, X, Y, Z).
mult(X, Y, Z) :- natrec(plus(X), Y, 0, Z).
step((X,Y),(Z,T)) :- succ(X,Z), mult(Z,Y,T).
factorial(X,Y) :- natrec(step,X,(0,1),(_,Y)).
Works fine, although eats quite some computing resources,
i.e. 9_303_219 inferences, since it must form
?- factorial(10,X).
X = 3628800.
?- time(factorial(10,X)).
% 9,303,219 inferences, 0.578 CPU in 0.617 seconds
(94% CPU, 16092054 Lips)
X = 3628800.

... fourier my ass, what has it to do with ordinals ...

Mild Shock

2024-03-01 20:08:51 UTC

I think the key are these terminological definitions:

"In order theory, a partial order is called well-founded if the
corresponding strict order is a well-founded relation. If the order is a
total order then it is called a well-order."
https://en.wikipedia.org/wiki/Well-founded_relation

So my tree "?" in question might have no infinite descend,
but it might not belong to the same total order, as the
other sets. But how exclude "?" ? The criteria of

transitive set is not violated:
trans(A) :<=> ∀ x , y : x ∈ A ∧ y ∈ x ⇒ y ∈ A
https://de.wikipedia.org/wiki/Transitive_Menge

One can easily verify that the above is satisfied by
the set "?". So what is violated? Well this
here is violate, namely each elememt should be

transitive as well, and so on:

"hereditarily transitive sets"
h-trans(A) :<=> trans(A) & ∀x(x ∈ A => h-trans(A))

Post by Mild Shock
? = o
/ | \
* o o
| |
* o
|
*
? = {{},{{}},{{{}}}}.

So I remember Jim Burns when he posited a more
general approach, he said transfinite induction
must be satisfied.

Otherwise we can take this Quine atom x = {x},
https://math.stackexchange.com/a/2874533

And by a suitable interpretation of the circular
h-trans definition, a definition that is not well-defined
since it has no unique interpretation,

we might judge this Quine atom an ordinal.

LoL

Post by Mild Shock
Know that one is the secret and source of all the cardinals.
-- Abraham ibn Ezra (1153)
But have mercy to me. So far I thought in ordinal
anlysis of programs, we would simply take the
tree of execution, and this is somehow an ordinal
for a terminating program? But whats the tree
repectively ordinal for for example 3! = 6 ?
Are all finite ordinals the same or not?
For example the ordinals 0, 1, 2, 3, with von
0 = {}
1 = {{}}
2 = {{},{{}}}
3 = {{},{{}},{{{},{{}}}}}.
0 =            *
1 =            o
               |
               *
2 =            o
              / \
             *   o
                 |
                 *
3 =            o
             / | \
           *   o    o
               |   / \
               * *   o
                      |
                      *
But what about this tree, it has also no infinite decend,
but what property is missing to make it an ordinal?
? =             o
              / | \
             * o o
                | |
                * o
                   |
                   *
? = {{},{{}},{{{}}}}.
Why is it not an ordinal?
P.S.: I tried to find an answer here, but I guess
I am too lazy to read it. Its starts with the funny
Trees, ordinals and termination
N Dershowitz · 1993
https://link.springer.com/content/pdf/10.1007/3-540-56610-4_68.pdf

... fourier my ass, what has it to do with ordinals ...

Jim Burns

2024-03-01 20:56:44 UTC

Post by Mild Shock
So I remember Jim Burns when he posited a more
general approach, he said transfinite induction
must be satisfied.

In a phrase I'm a little proud of, I said
transfinite.induction is well.order in drag.
One is a simple re-write of the other.

Post by Mild Shock
Otherwise we can take this Quine atom x = {x},
https://math.stackexchange.com/a/2874533
And by a suitable interpretation of the circular
h-trans definition, a definition that is not well-defined
since it has no unique interpretation,
we might judge this Quine atom an ordinal.
LoL

Ah.
But an ordinal is
a _regular_ transitive set of transitive sets.
So, not x = {x}

A regular non.empty set A holds
a disjoint element B.
∃B ∈ A: A∩B = ∅

But, if A is transitive.transitive,
each element is a subset, and
∀B ∈ A: A∩B = B
Transitive.transitive A can only be regular
if one of its elements is 0

∅ ∈ A ∧ ∅ ∈ A′ ties all the ordinals together.

It's a beautiful thing.

Mild Shock

2024-03-01 21:53:52 UTC

Lets work without regularity axiom, and
examine this naive attempt, hereditary =
my ancestors satisfied it as well:

"hereditarily transitive sets"
h-trans(A) :<=> trans(A) & ∀x(x ∈ A => h-trans(A))

Otherwise when regularity is present,
this excludes Quine atom q = {q}. When regularty is
not present, we can prove:

~h-trans(q)

Post by Mild Shock
So I remember Jim Burns when he posited a more
general approach, he said transfinite induction
must be satisfied.

In a phrase I'm a little proud of, I said
transfinite.induction is well.order in drag.
One is a simple re-write of the other.

Ah.
But an ordinal is
a _regular_ transitive set of transitive sets.
So, not x = {x}
A regular non.empty set A holds
a disjoint element B.
∃B ∈ A: A∩B = ∅
But, if A is transitive.transitive,
each element is a subset, and
∀B ∈ A: A∩B = B
Transitive.transitive A can only be regular
if one of its elements is 0
∅ ∈ A ∧ ∅ ∈ A′ ties all the ordinals together.
It's a beautiful thing.

Mild Shock

2024-03-01 22:11:36 UTC

Corr.: we cannot prove:

~h-trans(q)

See also the remark here by Andrés E. Caicedo:

Note that in the absence of foundation (= regularity),
this is a bit peculiar. For instance, if x={x}, then x
is hereditarily finite, although it does not belong to Vω.)
https://math.stackexchange.com/a/2874533

About digging into "transfinite.induction is well.order in drag"
by Jim Burns. You probably mean transfinite.induction
follows from well.order. What about the other direction?

Now my question, is assume we have no foundation,
but epsilon induction, what will happen. epsilon
induction is usually not an axiom. But what

if we stipulate it as an axiom?

Considered as an axiomatic principle, it is
called the axiom schema of set induction.
∀ x . ( ( ∀ ( y ∈ x ) . ψ ( y ) ) → ψ ( x ) ) → ∀ z . ψ ( z )
https://en.wikipedia.org/wiki/Epsilon-induction

Post by Mild Shock
Lets work without regularity axiom, and
examine this naive attempt, hereditary =
"hereditarily transitive sets"
h-trans(A) :<=> trans(A) & ∀x(x ∈ A => h-trans(A))
Otherwise when regularity is present,
this excludes Quine atom q = {q}. When regularty is
~h-trans(q)

Post by Mild Shock
So I remember Jim Burns when he posited a more
general approach, he said transfinite induction
must be satisfied.

In a phrase I'm a little proud of, I said
transfinite.induction is well.order in drag.
One is a simple re-write of the other.

Ah.
But an ordinal is
a _regular_ transitive set of transitive sets.
So, not x = {x}
A regular non.empty set A holds
a disjoint element B.
∃B ∈ A: A∩B = ∅
But, if A is transitive.transitive,
each element is a subset, and
∀B ∈ A: A∩B = B
Transitive.transitive A can only be regular
if one of its elements is 0
∅ ∈ A ∧ ∅ ∈ A′ ties all the ordinals together.
It's a beautiful thing.

Mild Shock

2024-03-01 22:33:18 UTC

My numb nut rewriting faculty, after spying
wikipeda, takes the contrapositive (i.e replace
A->B by ~B -> ~A) of the set induction axiom:

¬ ∀ z ψ ( z ) → ¬ ∀ x ( ( ∀ y ( y ∈ x → ψ ( y ) ) → ψ ( x ) )

Now use for ψ ( z ) the formula ¬ z ∈ u, and one gets:

∃ z z ∈ u → ∃ x ¬ ( ∀ y ( y ∈ x → ¬ y ∈ u ) → ¬ x ∈ u )

Again some contrapositive:

∃ z z ∈ u → ∃ x ¬ ( x ∈ u → ¬ ∀ y ( y ∈ x → ¬ y ∈ u ))

And hence:

∃ z z ∈ u → ∃ x ( x ∈ u ∧ ∀ y ( y ∈ x → ¬ y ∈ u ))

Some last quantifier switch, and we got the regularity axiom:

∃ z z ∈ u → ∃ x ( x ∈ u ∧ ∃ y ( y ∈ x ∧ y ∈ u ))

Usually written as:

u ≠ ∅ → ∃ x (x ∈ u ∧ x ∩ u ≠ ∅)

Post by Mild Shock
~h-trans(q)
Note that in the absence of foundation (= regularity),
this is a bit peculiar. For instance, if x={x}, then x
is hereditarily finite, although it does not belong to Vω.)
https://math.stackexchange.com/a/2874533
About digging into "transfinite.induction is well.order in drag"
by Jim Burns. You probably mean transfinite.induction
follows from well.order. What about the other direction?
Now my question, is assume we have no foundation,
but epsilon induction, what will happen. epsilon
induction is usually not an axiom. But what
if we stipulate it as an axiom?
Considered as an axiomatic principle, it is
called the axiom schema of set induction.
∀ x . ( ( ∀ ( y ∈ x ) . ψ ( y ) ) → ψ ( x ) ) → ∀ z . ψ ( z )
https://en.wikipedia.org/wiki/Epsilon-induction

Post by Mild Shock
So I remember Jim Burns when he posited a more
general approach, he said transfinite induction
must be satisfied.

In a phrase I'm a little proud of, I said
transfinite.induction is well.order in drag.
One is a simple re-write of the other.

Ah.
But an ordinal is
a _regular_ transitive set of transitive sets.
So, not x = {x}
A regular non.empty set A holds
a disjoint element B.
∃B ∈ A: A∩B = ∅
But, if A is transitive.transitive,
each element is a subset, and
∀B ∈ A: A∩B = B
Transitive.transitive A can only be regular
if one of its elements is 0
∅ ∈ A ∧ ∅ ∈ A′ ties all the ordinals together.
It's a beautiful thing.

Jim Burns

2024-03-02 00:41:34 UTC

Post by Mild Shock
About digging into
"transfinite.induction is well.order in drag"
by Jim Burns. You probably mean
transfinite.induction follows from well.order.
What about the other direction?

Both directions.

well.order

∃ᵒʳᵈγ:p(γ) ⟹ ∃#1ᵒʳᵈβ:p(β)

∃ᵒʳᵈγ:p(γ) ⟹ ∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α))

¬∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α)) ⟹ ¬∃ᵒʳᵈγ:p(γ)

∀ᵒʳᵈβ:(¬p(β) ∨ ¬∀ᵒʳᵈα<β:¬p(α)) ⟹ ∀ᵒʳᵈγ:¬p(γ)

∀ᵒʳᵈβ:(̅p(β) ∨ ¬∀ᵒʳᵈα<β:̅p(α)) ⟹ ∀ᵒʳᵈγ:̅p(γ)

∀ᵒʳᵈβ:(∀ᵒʳᵈα<β:̅p(α) ⇒ ̅p(β)) ⟹ ∀ᵒʳᵈγ:̅p(γ)

∀ᵒʳᵈβ:(̅p[0,β) ⇒ ̅p(β)) ⟹ ∀ᵒʳᵈγ:̅p(γ)

∀ᵒʳᵈβ:(̅p[0,β) ⇒ ̅p[0,β)∧̅p(β)) ⟹ ∀ᵒʳᵈγ:̅p(γ)

∀ᵒʳᵈβ:(̅p[0,β) ⇒ ̅p[0,β⁺¹)) ⟹ ∀ᵒʳᵈγ:̅p(γ)

transfinite.induction

Post by Mild Shock
Now my question, is assume we have no foundation,
but epsilon induction, what will happen. epsilon
induction is usually not an axiom. But what
if we stipulate it as an axiom?
Considered as an axiomatic principle, it is
called the axiom schema of set induction.
∀x.((∀(y∈x).ψ(y))→ψ(x))→∀z.ψ(z)
https://en.wikipedia.org/wiki/Epsilon-induction

That wiki.page assumes regular sets.
I'm not sure it does so explicitly,
but it defines ordinals as
transitive sets of transitive sets
which, without regularity, include x = {x}

Ross Finlayson

2024-03-02 04:15:25 UTC

Post by Mild Shock
About digging into
"transfinite.induction is well.order in drag"
by Jim Burns. You probably mean
transfinite.induction follows from well.order.
What about the other direction?

Both directions.
well.order
∃ᵒʳᵈγ:p(γ) ⟹ ∃#1ᵒʳᵈβ:p(β)
∃ᵒʳᵈγ:p(γ) ⟹ ∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α))
¬∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α)) ⟹ ¬∃ᵒʳᵈγ:p(γ)
∀ᵒʳᵈβ:(¬p(β) ∨ ¬∀ᵒʳᵈα<β:¬p(α)) ⟹ ∀ᵒʳᵈγ:¬p(γ)
∀ᵒʳᵈβ:(̅p(β) ∨ ¬∀ᵒʳᵈα<β:̅p(α)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
∀ᵒʳᵈβ:(∀ᵒʳᵈα<β:̅p(α) ⇒ ̅p(β)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
∀ᵒʳᵈβ:(̅p[0,β) ⇒ ̅p(β)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
∀ᵒʳᵈβ:(̅p[0,β) ⇒ ̅p[0,β)∧̅p(β)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
∀ᵒʳᵈβ:(̅p[0,β) ⇒ ̅p[0,β⁺¹)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
transfinite.induction

That wiki.page assumes regular sets.
I'm not sure it does so explicitly,
but it defines ordinals as
transitive sets of transitive sets
which, without regularity, include x = {x}

Like a process in time with no beginning?

The other day James Webb Space Telescope,
roundly paintcanned inflationary cosmology,
yet, even before that, the sky survey was measuring
the age of the universe every few years, and every
few years, it got hundreds of millions of years older.

Then the JWST sort of arrived at "you know it
really looks like we might have to start counting
over".

"Epoch".

A complementary notion to "Big Bang cosmology",
is, "Steady State cosmology", or, sitting next to
"Big Bang cosmology", "cyclic cosmology".

Two principles of theories of physics include
the dichotomy of unitarity and complementarity,
which is funny because one is without dichotomy
and the other is with.

2020/1/1 - 18262 = 1970/1/1

Time, then, reflects upon these foundational theories,
and anti-foundational theories, in simile, to Big Bang
theory, and Steady State theory.

Now, since scientism and logical positivism, Compte
and Boole and the Vienna circle, and Zermelo Fraenkel
and Le Maitre, with ZF set theory and Big Bang theory,
it was exactly about a century ago. 1920: a century
of hindsight, retrospect, from 2020.

So, the idea for delta-epsilonics toward zero, but
not crossing it, and least upper bound, then relying
on symmetry, sort of has the symmetry about the
origin exists for the symmetry about the origin to
exist.

Before DesCartes, one might aver the Euclid's theory,
geometry, was a bit free-er, geometry: do it anywhere
you want. Then, the notion of equipping the Euclidean
space, with a Cartesian space, makes exactly for the
notion of the ordinate itself, the ordinates and the
abscissae, which run or drop from the origin its axes,
to the curve its intercept, in these discussions of ordinals,
see arrive the notion of the ordinate, and the co-ordinates,
and that in all our real analysis, it's always based on the
co-ordinates.

So, how can there be negative numbers when first
the numbers must go all the way to infinity, if they
never reach it? The usual idea is that the comprehension
just goes ... around.

Then for DesCartes that introduces his notions of the
vortex everywhere, as anticipating particle/wave duality
with an implicit atomism, and for example: the spiral,
space-filling curve, an aspect of a: continuum.

One might imagine a faithful as possible simulacrum
of Einstein, a Zweistein: "I'm not making you say
the universe is infinite, but Space-Time is in a
continuous manifold".

So, unitarity and complementarity, the principles of
physics, sort of have the same notions in mathematics.

Ordinals: meet ordinates.

Ross Finlayson

2024-03-02 18:49:29 UTC

Post by Mild Shock
About digging into
"transfinite.induction is well.order in drag"
by Jim Burns. You probably mean
transfinite.induction follows from well.order.
What about the other direction?

Both directions.
well.order
∃ᵒʳᵈγ:p(γ) ⟹ ∃#1ᵒʳᵈβ:p(β)
∃ᵒʳᵈγ:p(γ) ⟹ ∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α))
¬∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α)) ⟹ ¬∃ᵒʳᵈγ:p(γ)
∀ᵒʳᵈβ:(¬p(β) ∨ ¬∀ᵒʳᵈα<β:¬p(α)) ⟹ ∀ᵒʳᵈγ:¬p(γ)
∀ᵒʳᵈβ:(̅p(β) ∨ ¬∀ᵒʳᵈα<β:̅p(α)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
∀ᵒʳᵈβ:(∀ᵒʳᵈα<β:̅p(α) ⇒ ̅p(β)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
∀ᵒʳᵈβ:(̅p[0,β) ⇒ ̅p(β)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
∀ᵒʳᵈβ:(̅p[0,β) ⇒ ̅p[0,β)∧̅p(β)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
∀ᵒʳᵈβ:(̅p[0,β) ⇒ ̅p[0,β⁺¹)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
transfinite.induction

That wiki.page assumes regular sets.
I'm not sure it does so explicitly,
but it defines ordinals as
transitive sets of transitive sets
which, without regularity, include x = {x}

Like a process in time with no beginning?
The other day James Webb Space Telescope,
roundly paintcanned inflationary cosmology,
yet, even before that, the sky survey was measuring
the age of the universe every few years, and every
few years, it got hundreds of millions of years older.
Then the JWST sort of arrived at "you know it
really looks like we might have to start counting
over".
"Epoch".
A complementary notion to "Big Bang cosmology",
is, "Steady State cosmology", or, sitting next to
"Big Bang cosmology", "cyclic cosmology".
Two principles of theories of physics include
the dichotomy of unitarity and complementarity,
which is funny because one is without dichotomy
and the other is with.
2020/1/1 - 18262 = 1970/1/1
Time, then, reflects upon these foundational theories,
and anti-foundational theories, in simile, to Big Bang
theory, and Steady State theory.
Now, since scientism and logical positivism, Compte
and Boole and the Vienna circle, and Zermelo Fraenkel
and Le Maitre, with ZF set theory and Big Bang theory,
it was exactly about a century ago. 1920: a century
of hindsight, retrospect, from 2020.
So, the idea for delta-epsilonics toward zero, but
not crossing it, and least upper bound, then relying
on symmetry, sort of has the symmetry about the
origin exists for the symmetry about the origin to
exist.
Before DesCartes, one might aver the Euclid's theory,
geometry, was a bit free-er, geometry: do it anywhere
you want. Then, the notion of equipping the Euclidean
space, with a Cartesian space, makes exactly for the
notion of the ordinate itself, the ordinates and the
abscissae, which run or drop from the origin its axes,
to the curve its intercept, in these discussions of ordinals,
see arrive the notion of the ordinate, and the co-ordinates,
and that in all our real analysis, it's always based on the
co-ordinates.
So, how can there be negative numbers when first
the numbers must go all the way to infinity, if they
never reach it? The usual idea is that the comprehension
just goes ... around.
Then for DesCartes that introduces his notions of the
vortex everywhere, as anticipating particle/wave duality
with an implicit atomism, and for example: the spiral,
space-filling curve, an aspect of a: continuum.
One might imagine a faithful as possible simulacrum
of Einstein, a Zweistein: "I'm not making you say
the universe is infinite, but Space-Time is in a
continuous manifold".
So, unitarity and complementarity, the principles of
physics, sort of have the same notions in mathematics.
Ordinals: meet ordinates.

Of course ordinals are about the slenderest,
leanest, most minimal sort of sets that establish
a course-of-passage, as what it's usually called
passing through ordinals, of infinite induction,
according to their structure, their form, their
content, their model.

For example they have almost none of the
modular character of the integers, though
each is different and they're ordered, it
involves counting back-and-forth and
up-and-down and building a mememory,
a structure the content the relation embodied,
form their scaffold a model, to relate a
model of _some_ ordinals, to a model
of _some_ integers.

So, in this way building in a usual way
models of abstract algebra's finite fields,
then on up, then on out, and for each of
those all theirs, sort of results that
_eventually_ then there's a huge structure
of all their intra-relation, "integers".

You might wonder that cardinals, in
set theory, maybe they're a little fuller?
A cardinal is an equivalence class of
_all the sets in the set-theoretic universe_
that in _all the models of functions among
those in the set-theoretic universe_ that
_all those functions indicating 1-1 and onto_
that those existing from a proto-typical
set with an element after zero's: equals
cardinal "1".

So, ordinals, basically got nothing, in
structure, except next, while cardinals,
are all the structure there is that in any
way relates anything at all a set, to a
set of prototypical elements, after
cardinal "0", as it were, cardinal "1".

Then, _ordinates_, are, way, way after that,
or for analytical geometers in their theory
kind of before, that way, way after there's
Euclidean geometry, then a Cartesian basis,
an origin and axes, with a metric on those
so it's a space and norm among those so
it's an orthonormal basis, the ordinates
are arrayed on out and down all those,
also implementing ordinals.

So, ordinals, meet ordinates, ordinates, ordinals.

Mild Shock

2024-03-02 19:40:45 UTC

Post by Mild Shock
Considered as an axiomatic principle, it is
called the axiom schema of set induction.
∀x.((∀(y∈x).ψ(y))→ψ(x))→∀z.ψ(z)
https://en.wikipedia.org/wiki/Epsilon-induction

That wiki.page assumes regular sets.

Well I showed classicaly that the set induction

Post by Jim Burns
My numb nut rewriting faculty, after spying
/* set induction */
¬ ∀ z ψ ( z ) → ¬ ∀ x ( ( ∀ y ( y ∈ x → ψ ( y ) ) → ψ ( x ) )
|
|
v
/* set induction */
u ≠ ∅ → ∃ x (x ∈ u ∧ x ∩ u ≠ ∅)

The only problem with this derivation, it
might not be intuitionistically valid. I might
have used a propositional or quantifier rules,

which are not accepted intuitionistically.

Ross Finlayson

2024-03-01 22:45:37 UTC

... fourier my ass, what has it to do with ordinals ...

Joseph Fourier? Fourier is famed for the Fourier-style
analysis, particularly the particular Fourier analysis.
Dirichlet and Fejer employ trigonometry.

One of it's most well-known counterparts is the
Taylor-style analysis, up after l'Hospitale's rule,
Rolle's theorem, up into Rodriguez formula.

The idea of putting them together about a
sort of Clairaut-MacLaurin for Fourier-Taylor,
about the zeros, is for that it's quite modern,
these sorts approaches.

When one mentions ordinals, as we've been
discussing ordering theory and the order-type
of ordinals and ORD the order-type of ordinals,
Cesare Burali-Forti's as it were, that the order,
the rulial, the regular, also lends to association,
the ordinance, or local laws, and the ordnance,
or, local laws.

So, when you say ORD, is it, loaded? The term?

Here it's got all the ordinals in it and contains itself.

If you didn't already know the theory of ordinals,
it's its own sort of primary element in the universe
of mathematical objects as for a theory by there being
a structure, a model theory, a model of same - ordinals,
and I suppose I've used the phrase "ubiquitous ordinals"
at least for twenty years.

It's a theory with numbers in it?

The ordering theory the axiomatic sub-field, has picked
up a lot of ground over the past decades, I don't much
know its particulars, except as with regards to that
the clock-arithmetic it makes for the modular, just like
the iota-values going zero to one, is about exactly how
it's done, here with regards usually enough to category
theory [0,1].

Some people use it instead of set theory,
it suffices for their needs.

Hey Burse, don't be counterfeiting. It's not just that
I don't like it, but governments and syndicates don't like it.
The law don't like it.

2024-03-01 08:47:23 UTC

Post by Richard Damon
And evvery natural number is finite and thus namable and thus visible.

That concerns potential infinity only.

Post by Richard Damon
It is the SET that "completes infinity", and the set isn't any of the
individual number.

The set is nothig but the collection of its elements. The complete set
requires that no element is missing. That proves, via ∀n ∈ ℕ: 1/n -
1/(n+1) = d_n > 0, the existence of a smallest unit fraction.

Regards, WM

Richard Damon

2024-03-01 14:44:40 UTC

Post by Richard Damon
And evvery natural number is finite and thus namable and thus visible.

That concerns potential infinity only.

No, it applies to ALL Finite numbers, which of course, being finte,
never actally REACH infinite, so if you want to invent a term for that
as "Potential Infinity", so be it.

But ALL

Post by Richard Damon
It is the SET that "completes infinity", and the set isn't any of the
individual number.

The set is nothig but the collection of its elements.

But it is, it is the COLLECTION of ALL its elements AT ONCE.

Thus, it converts the potential infinity of the individal Natural
Numbers, none of which are themselves infinite, into the ACTUAL infinity
of the set itself, and its size.

So, I guess your "dark" numbers, the ones that can only be talked about
as sets, are to collectively is the final collection (and its size).

Thus, your "NUF(x)" function is "dark" and you can't use its values
individually, so NUF(x) = 1 is an invalid statement.

Post by WM
The complete set
requires that no element is missing. That proves, via ∀n ∈ ℕ: 1/n -
1/(n+1) = d_n > 0, the existence of a smallest unit fraction.

Why does it require a smallest unit fraction?

Your equation proves to opposite.

Given that for all n 1/n - 1/(n+1) > 0

we can say that for all n (by addin 1/(n+1)

1/n > 1/(n+1)

Since for all n, n+1 exists (from the definition of the natural numbers)

Thus, there can not be a smallest 1/n

Smallest here only applies to finite sub-sets, not the final actually
infinite set.

Post by WM
Regards, WM

2024-03-01 18:29:13 UTC

Post by Richard Damon
And evvery natural number is finite and thus namable and thus visible.

That concerns potential infinity only.

No, it applies to ALL Finite numbers, which of course, being finte,
never actally REACH infinite, so if you want to invent a term for that
as "Potential Infinity", so be it.

But here we assume finished infinity.

Post by Richard Damon
It is the SET that "completes infinity", and the set isn't any of the
individual number.

The set is nothig but the collection of its elements.

But it is, it is the COLLECTION of ALL its elements AT ONCE.

There is nothing at once. Numbers are for counting one after the other.
Your claim shows that you need matheologial magic. But that's not maths.

Regards, WM

Richard Damon

2024-03-01 19:00:29 UTC

Post by Richard Damon
And evvery natural number is finite and thus namable and thus visible.

That concerns potential infinity only.

No, it applies to ALL Finite numbers, which of course, being finte,
never actally REACH infinite, so if you want to invent a term for that
as "Potential Infinity", so be it.

But here we assume finished infinity.

If we haven't finished the Infinity, you can't have you NUF(x), since it
starts at the infinite end.

Post by Richard Damon
It is the SET that "completes infinity", and the set isn't any of
the individual number.

The set is nothig but the collection of its elements.

But it is, it is the COLLECTION of ALL its elements AT ONCE.

There is nothing at once. Numbers are for counting one after the other.
Your claim shows that you need matheologial magic. But that's not maths.

And it seems your definition of "Maths" doesn't support the Natural Numbers.

But the actual Maths does, so you are using a defective version of logic
to think about it.

IF you can't support "All at Once" then you can't have the Set of
Natural Numbers to talk about, and thus we can't have your NUF.

You also can show that Achilles can't pass the Tortoise, as that
requires adding up the values "all at once" to let him catch up.

In other words, your logic is just insufficent for the task you are
using it for.

Post by WM
Regards, WM

2024-03-02 12:36:41 UTC

Post by Richard Damon
And evvery natural number is finite and thus namable and thus visible.

That concerns potential infinity only.

No, it applies to ALL Finite numbers, which of course, being finte,
never actally REACH infinite, so if you want to invent a term for that
as "Potential Infinity", so be it.

But here we assume finished infinity.

If we haven't finished the Infinity, you can't have you NUF(x), since it
starts at the infinite end.

True. NUF requires completed infinity. That is the premise.

Post by Richard Damon
IF you can't support "All at Once" then you can't have the Set of
Natural Numbers to talk about, and thus we can't have your NUF.
You also can show that Achilles can't pass the Tortoise, as that
requires adding up the values "all at once" to let him catch up.

Every step in half time is enough.

Regards, WM

Richard Damon

2024-03-02 14:22:48 UTC

Post by Richard Damon
And evvery natural number is finite and thus namable and thus visible.

That concerns potential infinity only.

No, it applies to ALL Finite numbers, which of course, being finte,
never actally REACH infinite, so if you want to invent a term for
that as "Potential Infinity", so be it.

But here we assume finished infinity.

If we haven't finished the Infinity, you can't have you NUF(x), since
it starts at the infinite end.

True. NUF requires completed infinity. That is the premise.

And thus must use a logic that ALLOWS for a completed infinity.

One at a time only doesn't.

Every step in half time is enough.

I guess you don't understand the problem.

Post by WM
Regards, WM

2024-03-02 15:25:41 UTC

Post by WM
NUF requires completed infinity. That is the premise.

And thus must use a logic that ALLOWS for a completed infinity.

Obviously there is none.

Post by Richard Damon
One at a time only doesn't.

One at a time only is the basis of counting. Using half the time for every
step would be possible. Otherwise there is no provable bijection. Only if
each desired step (not all) can be verified, it is matematics.

Regards, WM

mitchr...@gmail.com

2024-02-21 01:21:06 UTC

Post by ***@gmail.com

Post by William Elliot
Does the set of all ordinals exist within ZF?

It's too big to be a set.

Hmmm ...
It's certainly not a set in ZFC.
I'm not sure if the "too big" criterion can be applied in ZF.
But how would you _define_ such a set?
Wouldn't the postulated existence of such a set fall victim to
Russell's paradox?
quasi

No set can contain itself.

This is stupid. How can a set not contain itself?

Mild Shock

2024-02-26 12:01:44 UTC

Post by Jim Burns
Ordinals are well.ordered.

Only those which can be specified.

No.
All of them are well.ordered.

How do you know?

In the same way that I know
that right.triangles have three corners.

Yes, you are right. I exaggerated. Having three corners is essential

for triangles. Being well-ordered is essential for ordinals. What I
meant is that we cannot follow the well-order into the dark realm. In
particular Peano ceases.

Post by WM
Regards, WM

Peter Percival

2014-04-19 09:55:48 UTC

Post by William Elliot
Does the set of all ordinals exist within ZF?

No. In ZF there is a sort of counterpart of NBG's classes: consider the
formula with one free variable x, phi(x), that says "x is an ordinal",
then phi(x) can be treated in some ways as the class (it's a proper
class, not a set) of all ordinals. Also, if you're working with
ordinals, you might prefer ZFC to ZF.

Peter Percival

2014-04-19 10:09:47 UTC

Post by Peter Percival

Post by William Elliot
Does the set of all ordinals exist within ZF?

No. In ZF there is a sort of counterpart of NBG's classes: consider the
formula with one free variable x, phi(x), that says "x is an ordinal",

"x is an ordinal" is (according to von Neumann)

"x is transitive" & (Uy,z in x)(y=x \/ y in x \/ x in y)

"x is transitive" is (Uy in x)(y subset x)

Post by Peter Percival
then phi(x) can be treated in some ways as the class (it's a proper
class, not a set) of all ordinals. Also, if you're working with
ordinals, you might prefer ZFC to ZF.

Jim Burns

2014-04-19 11:18:23 UTC

Post by William Elliot
Does the set of all ordinals exist within ZF?

consider the formula with one free variable x, phi(x),
that says "x is an ordinal", then phi(x) can be treated
in some ways as the class (it's a proper class, not a set)
of all ordinals.

What is it that distinguishes sets from formulae?

Set notation, of course -- but that seems unimportant
and easily fixable without changing anything fundamental
in ZF.

And also sets can be elements of sets or proper classes.
What distinguishes "being an element of this set" from
"satisfying this formula"?

Speculating here, sets can act like both a formula itself
and like an object of a formula, of a variable within a
formula, the same set at the same time -- by which I mean,
contain and be contained.

This level-jumping can be done without sets, Goedel showed
us that, but sets do it promiscuously.

Also, if you're working with ordinals,
you might prefer ZFC to ZF.

Why would that be? Aren't ordinals well-ordered?

I would expect (without much thought) that ZF would be
fine since, if we are restricting our attention to
ordinals, we already have Choice as a theorem, call it
Choice Over Ordinals.

Peter Percival

2014-04-19 11:26:29 UTC

Post by William Elliot
Does the set of all ordinals exist within ZF?

consider the formula with one free variable x, phi(x),
that says "x is an ordinal", then phi(x) can be treated
in some ways as the class (it's a proper class, not a set)
of all ordinals.

What is it that distinguishes sets from formulae?
Set notation, of course -- but that seems unimportant
and easily fixable without changing anything fundamental
in ZF.
And also sets can be elements of sets or proper classes.
What distinguishes "being an element of this set" from
"satisfying this formula"?
Speculating here, sets can act like both a formula itself
and like an object of a formula, of a variable within a
formula, the same set at the same time -- by which I mean,
contain and be contained.
This level-jumping can be done without sets, Goedel showed
us that, but sets do it promiscuously.

Also, if you're working with ordinals,
you might prefer ZFC to ZF.

Why would that be? Aren't ordinals well-ordered?

Yes. I was just thinking that assuming choice might simplify things.

Post by Jim Burns
I would expect (without much thought) that ZF would be
fine since, if we are restricting our attention to
ordinals, we already have Choice as a theorem, call it
Choice Over Ordinals.

William Elliot

2014-04-20 02:27:09 UTC

Post by Peter Percival

Post by William Elliot
Does the set of all ordinals exist within ZF?

No. In ZF there is a sort of counterpart of NBG's classes: consider the
formula with one free variable x, phi(x), that says "x is an ordinal", then
phi(x) can be treated in some ways as the class (it's a proper class, not a
set) of all ordinals.

By transfinite induction is it possible (without AxC) to define
a set A_xi for every ordinal xi? With that can one proclaim
by replacement, that \/{ A_xi | xi an ordinal } is a set?

Post by Peter Percival
Also, if you're working with ordinals, you might prefer ZFC to ZF.

Not if you're proving propositions equivalent to AxC.

Peter Percival

2014-04-20 07:12:55 UTC

Post by William Elliot

Post by Peter Percival

Post by William Elliot
Does the set of all ordinals exist within ZF?

No. In ZF there is a sort of counterpart of NBG's classes: consider the
formula with one free variable x, phi(x), that says "x is an ordinal", then
phi(x) can be treated in some ways as the class (it's a proper class, not a
set) of all ordinals.

By transfinite induction is it possible (without AxC) to define
a set A_xi for every ordinal xi? With that can one proclaim

Yes, in ZF every ordinal is a set, so let A_xi be xi.

Post by William Elliot
by replacement, that \/{ A_xi | xi an ordinal } is a set?

Post by Peter Percival
Also, if you're working with ordinals, you might prefer ZFC to ZF.

Not if you're proving propositions equivalent to AxC.

zuhair

2014-04-19 17:46:06 UTC

Ross A. Finlayson

2014-04-19 19:18:08 UTC

Post by William Elliot
Does the set of all ordinals exist within ZF?

This is "Ord", a collection of all ordinals (from among their
representations). The paradox of Cesare Burali-Forti is that
structurally, where membership is used to model order, the
collection itself of the ordinals would be an ordinal, thus
including itself. A "paradox" is not a set in ZF.

Then there are set theories where it is a set, but those set
theories have anti-foundational infinities as a natural consequence
of definition. Russell has these kinds of sets as "extra-ordinary"
for ordinary.

foundational / anti-foundational
regular / irregular
well-founded / non-well-founded
ordinary / extra-ordinary

These are about the same.

There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.

ZF defines omega as a constant thus that omega and its products are
well-founded.

Mild Shock

2024-02-19 20:14:21 UTC

Whats the strategy for writing such nonsense as below?

What are products of omega? How are paradoxes sets?

LoL

Post by William Elliot
Does the set of all ordinals exist within ZF?

This is "Ord", a collection of all ordinals (from among their
representations). The paradox of Cesare Burali-Forti is that
structurally, where membership is used to model order, the
collection itself of the ordinals would be an ordinal, thus
including itself. A "paradox" is not a set in ZF.
Then there are set theories where it is a set, but those set
theories have anti-foundational infinities as a natural consequence
of definition. Russell has these kinds of sets as "extra-ordinary"
for ordinary.
foundational / anti-foundational
regular / irregular
well-founded / non-well-founded
ordinary / extra-ordinary
These are about the same.
There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.
ZF defines omega as a constant thus that omega and its products are
well-founded.

Mild Shock

2024-02-19 20:27:41 UTC

BTW: The finite ordinals as a set is omega. Omega does not
contain omega. Now it becomes clear what the strategy of
Rossy Boy is to write such nonsense.

Its a form of trolling based on:

Drunken Fist

Zui Quan

Drunken boxing (Chinese: 醉拳; pinyin: zuì quán) also
known as Drunken Fist, is a general name for all styles of
Chinese martial arts that imitate the movements of a
drunk person.[1] It is an ancient style and its origins are
mainly traced back to the Buddhist and Daoist religious
communities. The Buddhist style is related to the Shaolin
temple while the Daoist style is based on the Daoist tale of
the drunken Eight Immortals. Zui quan has the most unusual
body movements among all styles of Chinese martial arts.
Hitting, grappling, locking, dodging, feinting, ground and
aerial fighting and all other sophisticated methods of
combat are incorporated.

Post by Mild Shock
Whats the strategy for writing such nonsense as below?
What are products of omega? How are paradoxes sets?
LoL

Post by William Elliot
Does the set of all ordinals exist within ZF?

This is "Ord", a collection of all ordinals (from among their
representations). The paradox of Cesare Burali-Forti is that
structurally, where membership is used to model order, the
collection itself of the ordinals would be an ordinal, thus
including itself. A "paradox" is not a set in ZF.
Then there are set theories where it is a set, but those set
theories have anti-foundational infinities as a natural consequence
of definition. Russell has these kinds of sets as "extra-ordinary"
for ordinary.
foundational / anti-foundational
regular / irregular
well-founded / non-well-founded
ordinary / extra-ordinary
These are about the same.
There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.
ZF defines omega as a constant thus that omega and its products are
well-founded.

Ross Finlayson

2024-02-19 22:03:09 UTC

Post by Mild Shock
Whats the strategy for writing such nonsense as below?

(That sort of mercurial doffed-and-donned presumed jocularity and
familiarity is about the shallowest, vainest, fakest poser's.
That sort of inconstancy isn't "making friends and influencing people",
it's "give 'em nothing to depend on and keep 'em guessing".
It's the most obvious sort of example of a "manipulator",
which is considered a particular variety of pathological.)

Try some sincerety sometime.

Post by Mild Shock
What are products of omega? How are paradoxes sets?
LoL

Post by William Elliot
Does the set of all ordinals exist within ZF?

This is "Ord", a collection of all ordinals (from among their
representations). The paradox of Cesare Burali-Forti is that
structurally, where membership is used to model order, the
collection itself of the ordinals would be an ordinal, thus
including itself. A "paradox" is not a set in ZF.
Then there are set theories where it is a set, but those set
theories have anti-foundational infinities as a natural consequence
of definition. Russell has these kinds of sets as "extra-ordinary"
for ordinary.
foundational / anti-foundational
regular / irregular
well-founded / non-well-founded
ordinary / extra-ordinary
These are about the same.
There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.
ZF defines omega as a constant thus that omega and its products are
well-founded.

You mean "Russell lied to you and you bought it",
"Russell's retro-thesis", "Russell's fools"?

ORD, is the order type of ordinals, it's among
maximal elements and fixed points and universals.

It's not non-sense indeed the opposite.

My slates for uncountability and paradox,
help itemize how ordinals and sets are together.
(In a theory sets for ordinal relation, uncountability,
then a theory of sets with universes, paradox.)

(There's a theory of "ubiquitous ordinals" among
all the primordial objects of mathematics a theory
of them.)

If you study Cohen's "Independence of the Continuum Hypothesis",
right about at the end he introduces a deft consequence of ordinals,
and leaves set theory open about the Continuum Hypothesis.

In case you missed it, ....

It's pure theory, all theory.

It's called foundations, maybe you want to know it.

"Conservation of truth", all there is to it.

Ross Finlayson

2024-02-19 23:04:32 UTC

Post by Mild Shock
Whats the strategy for writing such nonsense as below?

Post by Mild Shock
What are products of omega? How are paradoxes sets?
LoL

Post by William Elliot
Does the set of all ordinals exist within ZF?

This is "Ord", a collection of all ordinals (from among their
representations). The paradox of Cesare Burali-Forti is that
structurally, where membership is used to model order, the
collection itself of the ordinals would be an ordinal, thus
including itself. A "paradox" is not a set in ZF.
Then there are set theories where it is a set, but those set
theories have anti-foundational infinities as a natural consequence
of definition. Russell has these kinds of sets as "extra-ordinary"
for ordinary.
foundational / anti-foundational
regular / irregular
well-founded / non-well-founded
ordinary / extra-ordinary
These are about the same.
There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.
ZF defines omega as a constant thus that omega and its products are
well-founded.

You mean "Russell lied to you and you bought it",
"Russell's retro-thesis", "Russell's fools"?
ORD, is the order type of ordinals, it's among
maximal elements and fixed points and universals.
It's not non-sense indeed the opposite.
My slates for uncountability and paradox,
help itemize how ordinals and sets are together.
(In a theory sets for ordinal relation, uncountability,
then a theory of sets with universes, paradox.)
(There's a theory of "ubiquitous ordinals" among
all the primordial objects of mathematics a theory
of them.)
If you study Cohen's "Independence of the Continuum Hypothesis",
right about at the end he introduces a deft consequence of ordinals,
and leaves set theory open about the Continuum Hypothesis.
In case you missed it, ....
It's pure theory, all theory.
It's called foundations, maybe you want to know it.
"Conservation of truth", all there is to it.

(Maybe that's just me.)

Mild Shock

2024-02-20 01:01:09 UTC

The contradiction is very easy:

Lets say X is the set of all finite ordinals.

- observe that X is an infinite ordinal.
- observe that if Y in X, then Y is a finite ordinal.
- hence if X in X it would be an infinite and finite ordinal at the same time.
- an X cannot be infinite and finite at the same time.

Post by Ross A. Finlayson
There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.

(Maybe that's just me.)

Ross Finlayson

2024-02-20 03:48:18 UTC

Post by Mild Shock
Lets say X is the set of all finite ordinals.
- observe that X is an infinite ordinal.
- observe that if Y in X, then Y is a finite ordinal.
- hence if X in X it would be an infinite and finite ordinal at the same time.
- an X cannot be infinite and finite at the same time.

Post by Ross A. Finlayson
There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.

(Maybe that's just me.)

Imagine if ordinals' proper model was that the successor
was powerset, instead of just any old ordered pair.

So, those together are the "sets that don't contain themselves",
the sets of ordinals.

Quantifying over those, results the "Russell set the ordinal",
it contains itself.

So here Y isn't necessarily a finite ordinal.

Q.E.R.

Mild Shock

2024-02-20 07:32:13 UTC

You only make it worse!

Post by Ross A. Finlayson
There are roundabout arguments that, for example,
the FINITE ORDINALS, as a set, consequently contain
themselves, as an element. This is a direct
compactness result.

If you want to have ordinals that contain themselves,
you need to mention an encoding. Because per se,
we understand by ordinal an order type.

There ware various encodings for finite ordinals around:
1) von Neuman encoding, based on succ(X) = X u {X} and 0 = {}
2) Zermelo encoding, bsaed on succ(X) = {X} and 0 = {}
3) Your Powerset idea, based on succ(X) = P(X) and 0 = {}

All 3 have the property that:

/* provable */
n in n+1 and n is finite

Proof:
case 1): n+1 = n u {n}, n in n+1 because n in {n}.
further succ(X) sendes an already finite set into a finite set.
case 2): n+1 = {n}, n in n+1 because n in {n}.
further succ(X) sendes an already finite set into a finite set.
case 3): n+1 = P(n), n in n+1 because n in P(n).
further succ(X) sendes an already finite set into a finite set.
Q.E.D.

But none has the property that omega = { n } contains
itself, the proof of contradiction applies irrelevant
of the encoding, it only makes use of the

notion finite and infinite:

/* provable */
~(omega in omega) & (Y in omega => Y finite)

Proof:
(Y in omega => Y finite) follows by the claim that
omega = { n }, i.e. the least set that contains all finite
ordinals in the corresponding encoding. If it would
contain something infinite it would not be the least

set that contains all finite ordinals, would have some
extra in it. Violating the very construction of omega from
the finite ordinals.
Q.E.D.

Post by Ross A. Finlayson
There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.

(Maybe that's just me.)

Imagine if ordinals' proper model was that the successor
was powerset, instead of just any old ordered pair.
So, those together are the "sets that don't contain themselves",
the sets of ordinals.
Quantifying over those, results the "Russell set the ordinal",
it contains itself.
So here Y isn't necessarily a finite ordinal.
Q.E.R.

Mild Shock

2024-02-20 07:57:36 UTC

You could use an encoding of finite ordinals
into infinite objects, like:

0 = omega, 1 = omega+1, etc..

Then my proof doesn't work so easily. You can then
use the regularity axiom, to show:

/* provable */
~(omega in omega)

Axiom of regularity
https://en.wikipedia.org/wiki/Axiom_of_regularity

Is this your A "paradox" is not a set in ZF?
In non-ZF you could aim at making omega a Quine atom:
https://en.wikipedia.org/wiki/Urelement#Quine_atoms

Or any other construction and encoding where you
would sneak in a set into itself.

Post by Mild Shock
You only make it worse!

Post by Ross A. Finlayson
There are roundabout arguments that, for example,
the FINITE ORDINALS, as a set, consequently contain
themselves, as an element. This is a direct
compactness result.

If you want to have ordinals that contain themselves,
you need to mention an encoding. Because per se,
we understand by ordinal an order type.
1) von Neuman encoding, based on succ(X) = X u {X} and 0 = {}
2) Zermelo encoding, bsaed on succ(X) = {X} and 0 = {}
3) Your Powerset idea, based on succ(X) = P(X) and 0 = {}
/* provable */
n in n+1 and n is finite
case 1): n+1 = n u {n}, n in n+1 because n in {n}.
further succ(X) sendes an already finite set into a finite set.
case 2): n+1 = {n}, n in n+1 because n in {n}.
further succ(X) sendes an already finite set into a finite set.
case 3): n+1 = P(n), n in n+1 because n in P(n).
further succ(X) sendes an already finite set into a finite set.
Q.E.D.
But none has the property that omega = { n } contains
itself, the proof of contradiction applies irrelevant
of the encoding, it only makes use of the
/* provable */
~(omega in omega) & (Y in omega => Y finite)
(Y in omega => Y finite) follows by the claim that
omega = { n }, i.e. the least set that contains all finite
ordinals in the corresponding encoding. If it would
contain something infinite it would not be the least
set that contains all finite ordinals, would have some
extra in it. Violating the very construction of omega from
the finite ordinals.
Q.E.D.

Post by Ross A. Finlayson
There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.

(Maybe that's just me.)

Imagine if ordinals' proper model was that the successor
was powerset, instead of just any old ordered pair.
So, those together are the "sets that don't contain themselves",
the sets of ordinals.
Quantifying over those, results the "Russell set the ordinal",
it contains itself.
So here Y isn't necessarily a finite ordinal.
Q.E.R.

Ross Finlayson

2024-02-20 17:53:30 UTC

Post by Mild Shock
You could use an encoding of finite ordinals
0 = omega, 1 = omega+1, etc..
Then my proof doesn't work so easily. You can then
/* provable */
~(omega in omega)
Axiom of regularity
https://en.wikipedia.org/wiki/Axiom_of_regularity
Is this your A "paradox" is not a set in ZF?
https://en.wikipedia.org/wiki/Urelement#Quine_atoms
Or any other construction and encoding where you
would sneak in a set into itself.

Post by Mild Shock
You only make it worse!

Post by Ross A. Finlayson
There are roundabout arguments that, for example,
the FINITE ORDINALS, as a set, consequently contain
themselves, as an element. This is a direct
compactness result.

Post by Ross A. Finlayson
There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.

(Maybe that's just me.)

Imagine if ordinals' proper model was that the successor
was powerset, instead of just any old ordered pair.
So, those together are the "sets that don't contain themselves",
the sets of ordinals.
Quantifying over those, results the "Russell set the ordinal",
it contains itself.
So here Y isn't necessarily a finite ordinal.
Q.E.R.

Thanks for writing, as you explore the issues involved with
quantification about ordinals and sets, it helps clarify
that "set theory" and "ordinal theory" are two different
theories, where being fundamental, each gets a very direct
model of the other in the respective theories.

Then, where "ordering theory" is about orderings, kind of
like category theory relating [0,1] to things as by functions,
that it's a function theory, ordering theory, here is that ordinal
theory is like set theory. There are "arithmetizations" of any thing
as there are "set-like associativities" of any thing, that's
the descriptive theory.

These kinds of ideas then get into that there is a theory of
mathematical objects, and these objects are same, whether
ordinals or sets (or parts, or differences, or otherwise
fundamental relations of utterly simple character that in
their classifications, effect relations, of other mathematical
objects of other mathematical object's theories, all in one theory.

So, when you look at something like Cohen's Independence of
the Continuum Hypothesis, it's very telling that it's a result
in ordinals, about cardinals, or here vice-versa.

You may be on the way to learning something.

Of course, the goal is "there are no paradoxes at all",
then what seem "inconsistent multiplicities", just don't relate.

(... That function theory effects "relations" that logic is
a theory of relations.)

Mild Shock

2024-02-20 19:10:48 UTC

Ordinals and Sets were developed hand in hand by Cantor
and Zermelo. But quasi, William Elliot, Peter Percival and
Jim Burns did already most of the explanations.

Post by Ross A. Finlayson
There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.

So you want to form a set of ordinals, the finite ones. Before
set theory ordinals were order types. This means they were
equivalence classes. Already the equivalence class of the

ordinal 1, is too big to be a set. Since it basically contains
all singleton sets {X}. And if you project the singleton you
get the universal class, which we know since Russell,

and even proved by Dan-O-Matik, isn't set.So you form
a collection of classes, sometimes called a conglomerate,
but you talk about it about as if it were a set.

So how can you make it a set? Well here is the receipt:

- Step 1: Start talking about numbers and transfinite numbers
Cantor 1895
- Step 2: Start mapping numbers [and transfinite numbers] to sets
Zermelo 1908

This was refined by von Neuman. Which gives the most useful
encoding of ordinals. Unless you want to go with Dana Scotts
trick. von Neuman ordinals not only have the property that

they are well ordered sets, their well ordering is the set
membership itself, they are hereditarily transitive sets.
You can construct inner models.

Post by Ross A. Finlayson
Thanks for writing, as you explore the issues involved with
quantification about ordinals and sets, it helps clarify
that "set theory" and "ordinal theory" are two different
theories, where being fundamental, each gets a very direct
model of the other in the respective theories.

Mild Shock

2024-02-20 19:23:42 UTC

To do some of Cohens work, you first have to accept
the Skolem Paradox, i.e. that ZF has countable models.

The Skolem Paradox is the thing that shattered shock
waves through Mückenheims brain, what does it do to

Rossy Boys brain? Oh, I forget Rossy Boy has no brain...

https://math.stackexchange.com/a/4027015

Post by Mild Shock
Ordinals and Sets were developed hand in hand by Cantor
and Zermelo. But quasi, William Elliot, Peter Percival and
Jim Burns did already most of the explanations.

Post by Ross A. Finlayson
There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.

So you want to form a set of ordinals, the finite ones. Before
set theory ordinals were order types. This means they were
equivalence classes. Already the equivalence class of the
ordinal 1, is too big to be a set. Since it basically contains
all singleton sets {X}. And if you project the singleton you
get the universal class, which we know since Russell,
and even proved by Dan-O-Matik, isn't set.So you form
a collection of classes, sometimes called a conglomerate,
but you talk about it about as if it were a set.
- Step 1: Start talking about numbers and transfinite numbers
Cantor 1895
- Step 2: Start mapping numbers [and transfinite numbers] to sets
Zermelo 1908
This was refined by von Neuman. Which gives the most useful
encoding of ordinals. Unless you want to go with Dana Scotts
trick. von Neuman ordinals not only have the property that
they are well ordered sets, their well ordering is the set
membership itself, they are hereditarily transitive sets.
You can construct inner models.

Ross Finlayson

2024-02-20 20:13:23 UTC

Post by Mild Shock
To do some of Cohens work, you first have to accept
the Skolem Paradox, i.e. that ZF has countable models.
The Skolem Paradox is the thing that shattered shock
waves through Mückenheims brain, what does it do to
Rossy Boys brain? Oh, I forget Rossy Boy has no brain...
https://math.stackexchange.com/a/4027015

Post by Mild Shock
Ordinals and Sets were developed hand in hand by Cantor
and Zermelo. But quasi, William Elliot, Peter Percival and
Jim Burns did already most of the explanations.

Post by Ross A. Finlayson
There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.

So you want to form a set of ordinals, the finite ones. Before
set theory ordinals were order types. This means they were
equivalence classes. Already the equivalence class of the
ordinal 1, is too big to be a set. Since it basically contains
all singleton sets {X}. And if you project the singleton you
get the universal class, which we know since Russell,
and even proved by Dan-O-Matik, isn't set.So you form
a collection of classes, sometimes called a conglomerate,
but you talk about it about as if it were a set.
- Step 1: Start talking about numbers and transfinite numbers
Cantor 1895
- Step 2: Start mapping numbers [and transfinite numbers] to sets
Zermelo 1908
This was refined by von Neuman. Which gives the most useful
encoding of ordinals. Unless you want to go with Dana Scotts
trick. von Neuman ordinals not only have the property that
they are well ordered sets, their well ordering is the set
membership itself, they are hereditarily transitive sets.
You can construct inner models.

Consider about "sets in ordinals" instead of
"ordinals in sets". There's a usual idea then
that ordinals besides reflecting an inductive set,
and a well-ordering in any mapping from them,
also that the whole numbers or "number theory's
numbers", so sit with them.

So, consider prime numbers with unique
factorization, and also a subset of prime
numbers that only have at most one power
of each prime factor. This is for the "fundamental
theorem of arithmetic", numbers (natural integers,
with regards to the cases for 0 and 1), that
numbers have unique prime factorizations,
and, of those, some have unique instances
of factors.

numbers: 1 2 3 4 5 6 ...
primes: 2 3 5 7 11 13 ...
composite: 4 6 8 9 10 12 14 15 ...
"uniposite": 2 3 5 6 7 10 11 13 ...

So, when modeling "sets", in "numbers",
there is a default model giving each element
in the universe of sets or the domain of discourse,
a prime number assignment, then a given set,
is just the multiple of those in these "uniposites",
where for example prime numbers generally
model "multi-sets", quite directly.

Now, I don't know too much who talks about
"primes and composites with only unique
factors in their decomposition, 'uniposites'",
but it would be interesting to really that a
very natural model of this sort _arithmetization_,
of sets, is exhibited _naturally_ because the
usual operation of union is taking the product
of the numbers and the usual operation of
membership is a divisibility test, and these kinds
of things.

union: product (least common multiple)
membership: divisibility test
intersection: greatest common denominator
disjoint: dividing out members

So, this introduces the usual notion of
"arithmetization", that then the operations
of arithmetic implement the set-theoretic
operations, then for such notions as
"geometrization", the sort of continuous analog,
of "arithmetization".

This sort of "composite numbers are a natural model
of a multi-set", can go a long way helping show that
the fundamental relations model and model and
model each other again, helping show why and how
it's simple that "resources in relations" establish
their orders, ..., of complexity in relation in type.

Here it's introduced the utility of
"uniposites: a sub-class of numbers
whose unique prime factorization has
unique elements".

2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47,...

https
oeis.org/search?q=2%2C3%2C5%2C6%2C7%2C10%2C11%2C13%2C14%2C15%2C17%2C19%2C21%2C23%2C26%2C29%2C31%2C33%2C34%2C35%2C37%2C39%2C41%2C42%2C43%2C46%2C47&language=english&go=Search

https
oeis.org/search?q=2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47&language=english&go=Search

Hmm, these aren't in the Online Encyclopedia of Integer Sequences?

https
oeis.org/A000040
https
oeis.org/A002808

The idea is simple that multi-sets are modeled by numbers ,
and, sets modeled by these "uniposite", numbers. Surely,
or rather, Shirley almost certainly, these already are,
"known".

A most usual sort of modeling a set as a number
is a "bit-map", for a word as wide as
the domain-of-discourse.

Ross Finlayson

2024-02-20 22:12:37 UTC

Post by Mild Shock
Ordinals and Sets were developed hand in hand by Cantor
and Zermelo. But quasi, William Elliot, Peter Percival and
Jim Burns did already most of the explanations.

Post by Ross A. Finlayson
There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.

So you want to form a set of ordinals, the finite ones. Before
set theory ordinals were order types. This means they were
equivalence classes. Already the equivalence class of the
ordinal 1, is too big to be a set. Since it basically contains
all singleton sets {X}. And if you project the singleton you
get the universal class, which we know since Russell,
and even proved by Dan-O-Matik, isn't set.So you form
a collection of classes, sometimes called a conglomerate,
but you talk about it about as if it were a set.
- Step 1: Start talking about numbers and transfinite numbers
Cantor 1895
- Step 2: Start mapping numbers [and transfinite numbers] to sets
Zermelo 1908
This was refined by von Neuman. Which gives the most useful
encoding of ordinals. Unless you want to go with Dana Scotts
trick. von Neuman ordinals not only have the property that
they are well ordered sets, their well ordering is the set
membership itself, they are hereditarily transitive sets.
You can construct inner models.

Consider about "sets in ordinals" instead of
"ordinals in sets". There's a usual idea then
that ordinals besides reflecting an inductive set,
and a well-ordering in any mapping from them,
also that the whole numbers or "number theory's
numbers", so sit with them.
So, consider prime numbers with unique
factorization, and also a subset of prime
numbers that only have at most one power
of each prime factor. This is for the "fundamental
theorem of arithmetic", numbers (natural integers,
with regards to the cases for 0 and 1), that
numbers have unique prime factorizations,
and, of those, some have unique instances
of factors.
numbers: 1 2 3 4 5 6 ...
primes: 2 3 5 7 11 13 ...
composite: 4 6 8 9 10 12 14 15 ...
"uniposite": 2 3 5 6 7 10 11 13 ...
So, when modeling "sets", in "numbers",
there is a default model giving each element
in the universe of sets or the domain of discourse,
a prime number assignment, then a given set,
is just the multiple of those in these "uniposites",
where for example prime numbers generally
model "multi-sets", quite directly.
Now, I don't know too much who talks about
"primes and composites with only unique
factors in their decomposition, 'uniposites'",
but it would be interesting to really that a
very natural model of this sort _arithmetization_,
of sets, is exhibited _naturally_ because the
usual operation of union is taking the product
of the numbers and the usual operation of
membership is a divisibility test, and these kinds
of things.
union: product (least common multiple)
membership: divisibility test
intersection: greatest common denominator
disjoint: dividing out members
So, this introduces the usual notion of
"arithmetization", that then the operations
of arithmetic implement the set-theoretic
operations, then for such notions as
"geometrization", the sort of continuous analog,
of "arithmetization".
This sort of "composite numbers are a natural model
of a multi-set", can go a long way helping show that
the fundamental relations model and model and
model each other again, helping show why and how
it's simple that "resources in relations" establish
their orders, ..., of complexity in relation in type.
Here it's introduced the utility of
"uniposites: a sub-class of numbers
whose unique prime factorization has
unique elements".
2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47,...
https
oeis.org/search?q=2%2C3%2C5%2C6%2C7%2C10%2C11%2C13%2C14%2C15%2C17%2C19%2C21%2C23%2C26%2C29%2C31%2C33%2C34%2C35%2C37%2C39%2C41%2C42%2C43%2C46%2C47&language=english&go=Search
https
oeis.org/search?q=2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47&language=english&go=Search
Hmm, these aren't in the Online Encyclopedia of Integer Sequences?
https
oeis.org/A000040
https
oeis.org/A002808
The idea is simple that multi-sets are modeled by numbers ,
and, sets modeled by these "uniposite", numbers. Surely,
or rather, Shirley almost certainly, these already are,
"known".
A most usual sort of modeling a set as a number
is a "bit-map", for a word as wide as
the domain-of-discourse.

Numbers with max(multiplicity(prime-factor)) = 1

Post by Ross Finlayson
Here it's introduced the utility of
"uniposites: a sub-class of numbers
whose unique prime factorization has
unique elements".

2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47,...

Post by Ross Finlayson
https

oeis.org/search?q=2%2C3%2C5%2C6%2C7%2C10%2C11%2C13%2C14%2C15%2C17%2C19%2C21%2C23%2C26%2C29%2C31%2C33%2C34%2C35%2C37%2C39%2C41%2C42%2C43%2C46%2C47&language=english&go=Search

Post by Ross Finlayson
https

oeis.org/search?q=2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47&language=english&go=Search

Post by Ross Finlayson
Hmm, these aren't in the Online Encyclopedia of Integer Sequences?
https
oeis.org/A000040
https
oeis.org/A002808

https
oeis.org/wiki/Prime_factors

"The arithmetic function omega(n) represents
the number of distinct prime factors of n

omega(n) = Sigma_i=1^omega(n) alpha_i^0 = Sigma_i=1^omega(n) 1

if you forgive the tautological expression."

https
en.wikipedia.org/wiki/Table_of_prime_factors

"A powerful number (also called squareful) has multiplicity
above 1 for all prime factors."

Ah, "square-free integers". "A square-free integer has no
prime factor with multiplicity above 1".

https
oeis.org/A005117

So, the square-free numbers are natural representatives of
subsets in arithmetic, of natural representatives of powersets
in arithmetic, where elements have natural representations
as prime numbers, and the set of all of them is called the
"primorial" which is like "factorial" for the first n-many primes.

(Here this is "square-free numbers excluding 1".)

2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47,...

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30,
31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47

Ah, looks I omitted 22 and 38, and 30.

Yes, of course square-free numbers are very well-known.

https
mathoverflow.net/questions/16098/complexity-of-testing-integer-square-freeness

It reminds me of "Digit Summation Congruence", which is
a method for machine numbers that rapidly tests divisibility
using binary arithmetic, for various prime factors.

Any sorts addition formula are usually considered
very conducive to tractability.

Ross Finlayson

2024-02-20 20:41:39 UTC

Post by Mild Shock
Ordinals and Sets were developed hand in hand by Cantor
and Zermelo. But quasi, William Elliot, Peter Percival and
Jim Burns did already most of the explanations.

Post by Ross A. Finlayson
There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.

So you want to form a set of ordinals, the finite ones. Before
set theory ordinals were order types. This means they were
equivalence classes. Already the equivalence class of the
ordinal 1, is too big to be a set. Since it basically contains
all singleton sets {X}. And if you project the singleton you
get the universal class, which we know since Russell,
and even proved by Dan-O-Matik, isn't set.So you form
a collection of classes, sometimes called a conglomerate,
but you talk about it about as if it were a set.
- Step 1: Start talking about numbers and transfinite numbers
Cantor 1895
- Step 2: Start mapping numbers [and transfinite numbers] to sets
Zermelo 1908
This was refined by von Neuman. Which gives the most useful
encoding of ordinals. Unless you want to go with Dana Scotts
trick. von Neuman ordinals not only have the property that
they are well ordered sets, their well ordering is the set
membership itself, they are hereditarily transitive sets.
You can construct inner models.

Actually the Paul Cohen's "Independence of the Continuum,
Hypothesis, 1 and 2" that I read was from the original as
I found a copy on the National Institute of Health's web-page,
so, I read it from there, instead of taking it second-hand
from a conservative crowd of reputation-mongers.

So, when you read it, at the very end, it's like,
"surprise: ordinal's bigger".

On sci.logic one time there's a thread called
"Few questions on forcing, large cardinals".

https
groups.google.com/g/sci.logic/c/sIvO0bJ7gPY/m/VBUICn3tBAAJ

It appears that what the Burse-a-tron emitted on 1/24/2020
was "That's quite amazing!".

So, anyways, about Cohen and "Independence of the Continuum
Hypothesis", it's not necessarily easy to find a copy, but,
you want it through your own lens.

Here's a few more through mine:

https
groups.google.com/g/sci.logic/search?q=Cohen%20author%3AFinlayson

There are a few more on sci.math, alzo.

Ross Finlayson

2024-12-26 19:18:39 UTC

Post by Mild Shock
Whats the strategy for writing such nonsense as below?

Post by Mild Shock
What are products of omega? How are paradoxes sets?
LoL