Discussion:
Is There an Official Answer to Division by Zero?
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Doug
2017-07-16 15:45:41 UTC
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Is this one of those unsolved mysteries of Mathematics, or is it just too high level for the average person? I tried to search but didn't find anything that looked like an "official" or commonly accepted answer.

Thanks!
Richard Tobin
2017-07-17 13:30:00 UTC
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Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just
too high level for the average person?
There isn't any mystery about it. You just can't do it.

For example, there isn't any number that gives you 4 when you multiply
it by 0, so 4 can't be divided by 0.

-- Richard
Peter Percival
2017-07-17 14:33:33 UTC
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Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just
too high level for the average person? I tried to search but didn't
find anything that looked like an "official" or commonly accepted
answer.
x/y = z means the same as x = zy. If y = 0 and x =/= 0 then x = zy has
no solutions z. If y = 0 and x = 0 then x = zy has any number z as a
solution. For these reasons x/0 is undefined whatever x.
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
FredJeffries
2017-07-17 15:16:47 UTC
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Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just too high level for the average person? I tried to search but didn't find anything that looked like an "official" or commonly accepted answer.
There is no one "official" answer because the answer you get depends on the consequences you are willing to live with. In the words of James Tanton, "Make it happen. Deal with the consequences."

There are several fruitful avenues to take in this area and probably more waiting to be discovered/invented. Which one(s) you decide to explore depends upon what exact questions you are trying to answer and what you are willing to give up. For the more popular approaches, search terms might be "projective line", "Riemann sphere", "renormalization"

https://en.wikipedia.org/wiki/Division_by_zero

I highly recommend David W Cantrell's "How I Divided by Zero, and Lived to Tell About It!"
http://mathforum.org/kb/message.jspa?messageID=312626
http://mathforum.org/kb/message.jspa?messageID=4643254
Markus Klyver
2017-07-17 15:36:51 UTC
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Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just too high level for the average person? I tried to search but didn't find anything that looked like an "official" or commonly accepted answer.
Thanks!
You can extend the reals and define a/0 := ∞ for every real number a.
bassam king karzeddin
2017-07-17 17:17:11 UTC
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Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just too high level for the average person? I tried to search but didn't find anything that looked like an "official" or commonly accepted answer.
Thanks!
Just remember that zero was not any real number or integer, it was purposely manufactured to facilitate the calculations for merchants that were too difficult to perform in Roman numbers

So, considering the reality of zero being a real integer was actually illegal, it was made so by little human definitions, but the reality is completely independent of our own little meaningless definitions

So, it is indeed the representation of absolute nothingness, contradicting the term of reality that had been illegally added to it

So, reconsidering it back as a symbol of not any real integer would immediately resolve all the nonsense volumes that had been aroused by it

Then it becomes more than clear that all operations conducted on reality are so meaningless for all those manufactured integers

But, this so, unfortunately, would sweep away most of your mathematics that you enjoy currently since it was based on many of such alike fictions

But truly, the absolute facts including mathematics are so independent of our definitions, (so, this is the real tragedy that we had placed ourselves inside and so deliberately for sure)

BKK
Vinicius Claudino Ferraz
2017-07-17 17:53:41 UTC
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zero is the anihilator of ℝ

x ∈ ℝ ⇒ 0 * x = 0

if you not assume that, it's not zero. It would be a fake_zero.
Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just too high level for the average person? I tried to search but didn't find anything that looked like an "official" or commonly accepted answer.
Thanks!
Markus Klyver
2017-07-17 21:35:38 UTC
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Post by Vinicius Claudino Ferraz
zero is the anihilator of ℝ
x ∈ ℝ ⇒ 0 * x = 0
if you not assume that, it's not zero. It would be a fake_zero.
Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just too high level for the average person? I tried to search but didn't find anything that looked like an "official" or commonly accepted answer.
Thanks!
What you mean is that 0 is the null element in the field of real numbers. Of course, because otherwise ℝ wouldn't be a field.
FredJeffries
2017-07-18 12:20:18 UTC
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Post by Markus Klyver
Post by Vinicius Claudino Ferraz
zero is the anihilator of ℝ
x ∈ ℝ ⇒ 0 * x = 0
if you not assume that, it's not zero. It would be a fake_zero.
Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just too high level for the average person? I tried to search but didn't find anything that looked like an "official" or commonly accepted answer.
Thanks!
What you mean is that 0 is the null element in the field of real numbers. Of course, because otherwise ℝ wouldn't be a field.
No he doesn't. He means that 0 is an absorbing element in the magma of real numbers.
https://en.wikipedia.org/wiki/Absorbing_element
Markus Klyver
2017-07-18 14:47:52 UTC
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Post by FredJeffries
Post by Markus Klyver
Post by Vinicius Claudino Ferraz
zero is the anihilator of ℝ
x ∈ ℝ ⇒ 0 * x = 0
if you not assume that, it's not zero. It would be a fake_zero.
Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just too high level for the average person? I tried to search but didn't find anything that looked like an "official" or commonly accepted answer.
Thanks!
What you mean is that 0 is the null element in the field of real numbers. Of course, because otherwise ℝ wouldn't be a field.
No he doesn't. He means that 0 is an absorbing element in the magma of real numbers.
https://en.wikipedia.org/wiki/Absorbing_element
...

Which is basically the same thing since the algebraic closures of a given field also form a groupoid, where all objects are isomophic. "The null element" and "the absorbing element" is usually used interchangeably and mean the same thing.
FredJeffries
2017-07-20 01:24:53 UTC
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Post by Markus Klyver
Post by FredJeffries
Post by Markus Klyver
Post by Vinicius Claudino Ferraz
zero is the anihilator of ℝ
x ∈ ℝ ⇒ 0 * x = 0
if you not assume that, it's not zero. It would be a fake_zero.
Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just too high level for the average person? I tried to search but didn't find anything that looked like an "official" or commonly accepted answer.
Thanks!
What you mean is that 0 is the null element in the field of real numbers. Of course, because otherwise ℝ wouldn't be a field.
No he doesn't. He means that 0 is an absorbing element in the magma of real numbers.
https://en.wikipedia.org/wiki/Absorbing_element
...
Which is basically the same thing since the algebraic closures of a given field also form a groupoid, where all objects are isomophic. "The null element" and "the absorbing element" is usually used interchangeably and mean the same thing.
Sorry. I had never seen "null element" used as a synonym for "absorbing element" (or annihilating element). I apologize for the misunderstanding.
Steve Carroll
2017-07-18 14:14:20 UTC
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Holy shit another mathforum.org troll.

"Doug" wrote in message news:***@sodium.mathforum.org...

Try using a "search engine", you drooling half-wit. I understand "teh
google" is vey popular these days with the kids.
Mathedman
2017-07-18 14:18:21 UTC
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Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just too high level for the average person? I tried to search but didn't find anything that looked like an "official" or commonly accepted answer.
Thanks!
The "official" (there is no "official" in mathematics) answer is
that division by zero is not a defined concept. No can there be ---
else muliplication by zero would not yield zero!
Markus Klyver
2017-07-18 14:49:41 UTC
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Post by Mathedman
Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just too high level for the average person? I tried to search but didn't find anything that looked like an "official" or commonly accepted answer.
Thanks!
The "official" (there is no "official" in mathematics) answer is
that division by zero is not a defined concept. No can there be ---
else muliplication by zero would not yield zero!
Of course you can define division by zero and still have x*0 = 0*x = 0. Stop talking nonsense.
Vinicius Claudino Ferraz
2017-07-19 16:42:06 UTC
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No. Let (G, x) a group under multiplication.

zero can't be an element of G, because: what is its inverse?

suppose 0^(-1) = i
1i = i
2i = 2 * 0^(-1) = y
y * 0 = 2
z * 0 = 3

0i = 0 * 0^(-1) = w
w * 0 = 0 ==> 0i is not unique.
Post by Markus Klyver
Of course you can define division by zero and still have x*0 = 0*x = 0. Stop talking nonsense.
Dan Christensen
2017-07-18 17:44:00 UTC
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Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just too high level for the average person? I tried to search but didn't find anything that looked like an "official" or commonly accepted answer.
It is only an "unsolved mystery" to cranks and trolls.

Division on the reals (or natural numbers, rationals, integers) is defined in terms of multiplication as follows:

For all x, y, z in R, if y=/=0 then x/y=z iff x=z*y.

Why the restriction y=/=0? Because x=z*y has infinitely many solutions for z when x=0, and none when x=/=0.

Dan

Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
Dan Christensen
2017-07-20 12:49:00 UTC
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(Correction)
Post by Dan Christensen
Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just too high level for the average person? I tried to search but didn't find anything that looked like an "official" or commonly accepted answer.
It is only an "unsolved mystery" to cranks and trolls.
For all x, y, z in R, if y=/=0 then x/y=z iff x=z*y.
Why the restriction y=/=0?
Suppose y=0. Then x=z*y has infinitely many solutions for z for x=0, and none for x=/=0. When y=0, there is never a unique solution for z.
Post by Dan Christensen
Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
Franz Gnaedinger
2017-07-20 07:23:46 UTC
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Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just too high level for the average person? I tried to search but didn't find anything that looked like an "official" or commonly accepted answer.
Thanks!
Mathematics, in my opinion, is the logic of building and maintaining,
based on the formula a = a while natural language follows the wider logic
of equal unequal inherent in life. Goethe: All is equal, all unequal ...
Dividing by zero yields infinite, which is equal unequal in itself, and thus
marks the border between mathematics and the wider logic of life. Paradoxa
like for example the liar paradox also mark this border. And Goedel proved
that there _is_ a beyond of mathematical logic. So far my interpretation,
not really (or not yet) the official version.
Doug
2017-07-20 05:15:14 UTC
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row row row your boat, life is but a dream
Doug
2017-07-20 05:13:03 UTC
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Thanks! I will check those out
Doug
2017-07-20 05:20:42 UTC
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*Some* of these answers were informative and thought provoking. Google was a massive waste of time.

Long long ago, in a different age of man, people would stop and ask for directions. It was quite unusual for the answer to be "go buy a map, you drooling half-wit" ;)
Doug
2017-07-20 06:08:56 UTC
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For some reason my posts take a long time to post, so I don't know if my other "thanks" is going to ever show up or not...

In any case, thanks again. I checked out the things you mentioned.

This has convinced me to write up my own "proof"/musings on the subject.

I know it sounds a little trollish, but I think the honest answer here is that it is a hole in our current understanding. The answer being either "impossible" or "can only be done by imaginary constructs that nobody understands" leads me to believe that we have simply failed to solve the problem :)

The Riemann sphere and other things you mentioned are interesting, at least they created some sort of placeholder or attempt to work with the issue. It just doesn't seem like they quite made it to the end of the rainbow, so to speak.

I remember back in my school days when this came up we were told that the simple answer is it can't be done, but that mathematicians have got it figured out, it's just too complicated for normal humans.
Julio Di Egidio
2017-07-20 13:37:40 UTC
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Post by Doug
I know it sounds a little trollish, but I think the honest answer here is that it is a hole in our current understanding. The answer being either "impossible" or "can only be done by imaginary constructs that nobody understands" leads me to believe that we have simply failed to solve the problem :)
As some have tried to explain, division by zero is defined in some systems,
not defined in others. There is no mystery about it, just insane bullshit
by the resident spammers.

HTH,

Julio
Dan Christensen
2017-07-21 04:00:20 UTC
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Post by Julio Di Egidio
Post by Doug
I know it sounds a little trollish, but I think the honest answer here is that it is a hole in our current understanding. The answer being either "impossible" or "can only be done by imaginary constructs that nobody understands" leads me to believe that we have simply failed to solve the problem :)
As some have tried to explain, division by zero is defined in some systems,
not defined in others.
You seem to be suggesting that these other "systems" are, in some sense, equivalent to that which is widely used, and that they are somehow equally useful in applications. That is pure nonsense, of course.

Dan
Julio Di Egidio
2017-07-21 04:44:09 UTC
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Post by Dan Christensen
Post by Julio Di Egidio
Post by Doug
I know it sounds a little trollish, but I think the honest answer here is that it is a hole in our current understanding. The answer being either "impossible" or "can only be done by imaginary constructs that nobody understands" leads me to believe that we have simply failed to solve the problem :)
As some have tried to explain, division by zero is defined in some systems,
not defined in others.
You seem to be suggesting that these other "systems" are, in some sense,
equivalent to that which is widely used,
Possibly to you who are stupid.

Julio
Peter Percival
2017-07-21 12:01:54 UTC
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Post by Dan Christensen
Post by Julio Di Egidio
Post by Doug
I know it sounds a little trollish, but I think the honest answer
here is that it is a hole in our current understanding. The
answer being either "impossible" or "can only be done by
imaginary constructs that nobody understands" leads me to believe
that we have simply failed to solve the problem :)
As some have tried to explain, division by zero is defined in some
systems, not defined in others.
You seem to be suggesting that these other "systems" are, in some
sense, equivalent to that which is widely used, and that they are
somehow equally useful in applications. That is pure nonsense, of
course.
Of course. No one in their right mind would want to have anything to do
with the Riemann sphere or projective geometry.
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
Dan Christensen
2017-07-21 23:06:22 UTC
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Post by Peter Percival
Post by Dan Christensen
Post by Julio Di Egidio
Post by Doug
I know it sounds a little trollish, but I think the honest answer
here is that it is a hole in our current understanding. The
answer being either "impossible" or "can only be done by
imaginary constructs that nobody understands" leads me to believe
that we have simply failed to solve the problem :)
As some have tried to explain, division by zero is defined in some
systems, not defined in others.
You seem to be suggesting that these other "systems" are, in some
sense, equivalent to that which is widely used, and that they are
somehow equally useful in applications. That is pure nonsense, of
course.
Of course. No one in their right mind would want to have anything to do
with the Riemann sphere or projective geometry.
The natural numbers, integers, rationals, reals and complex numbers can all apparently be constructed in ZFC theory. What about the extension of the reals to "infinity?"

Dan

Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
Shobe, Martin
2017-07-22 00:35:03 UTC
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Post by Dan Christensen
Post by Peter Percival
Post by Dan Christensen
Post by Julio Di Egidio
Post by Doug
I know it sounds a little trollish, but I think the honest answer
here is that it is a hole in our current understanding. The
answer being either "impossible" or "can only be done by
imaginary constructs that nobody understands" leads me to believe
that we have simply failed to solve the problem :)
As some have tried to explain, division by zero is defined in some
systems, not defined in others.
You seem to be suggesting that these other "systems" are, in some
sense, equivalent to that which is widely used, and that they are
somehow equally useful in applications. That is pure nonsense, of
course.
Of course. No one in their right mind would want to have anything to do
with the Riemann sphere or projective geometry.
The natural numbers, integers, rationals, reals and complex numbers can all apparently be constructed in ZFC theory. What about the extension of the reals to "infinity?"
All the ones I know of can.

Martin Shobe
Dan Christensen
2017-07-22 01:04:05 UTC
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Post by Shobe, Martin
Post by Dan Christensen
Post by Peter Percival
Post by Dan Christensen
Post by Julio Di Egidio
Post by Doug
I know it sounds a little trollish, but I think the honest answer
here is that it is a hole in our current understanding. The
answer being either "impossible" or "can only be done by
imaginary constructs that nobody understands" leads me to believe
that we have simply failed to solve the problem :)
As some have tried to explain, division by zero is defined in some
systems, not defined in others.
You seem to be suggesting that these other "systems" are, in some
sense, equivalent to that which is widely used, and that they are
somehow equally useful in applications. That is pure nonsense, of
course.
Of course. No one in their right mind would want to have anything to do
with the Riemann sphere or projective geometry.
The natural numbers, integers, rationals, reals and complex numbers can all apparently be constructed in ZFC theory. What about the extension of the reals to "infinity?"
All the ones I know of can.
Given the real numbers, what axioms of ZFC would you use to extend them to include an infinite number?

Dan

Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
FredJeffries
2017-07-22 01:31:35 UTC
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Post by Dan Christensen
Given the real numbers, what axioms of ZFC would you use to extend them to include an infinite number?
Axiom of Union
Dan Christensen
2017-07-22 01:45:01 UTC
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Post by FredJeffries
Post by Dan Christensen
Given the real numbers, what axioms of ZFC would you use to extend them to include an infinite number?
Axiom of Union
The union of what set(s)? Using one-sided Dedekind cuts, the union of the set of real numbers is just the set of rational numbers. Yes, the set of rational numbers is infinite, but it would be a stretch to call it the extension of the reals to "infinity."

Dan

Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
Jim Burns
2017-07-22 02:13:32 UTC
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On Friday, July 21, 2017 at 9:31:47 PM UTC-4,
On Friday, July 21, 2017 at 6:04:12 PM UTC-7,
Post by Dan Christensen
Given the real numbers, what axioms of ZFC would you
use to extend them to include an infinite number?
Axiom of Union
The union of what set(s)?
Using one-sided Dedekind cuts,
the union of the set of real numbers is just
the set of rational numbers.
Yes, the set of rational numbers is infinite,
That's a different kind of infinity, cardinality.
but it would be a stretch to call it the extension
of the reals to "infinity."
Recall that you gave the real numbers.

For each real number x (of the unextended reals),
there is exactly one set of real numbers X = {y| y < x}
the set of reals y less than x.

Define an injection L: R -> P(R) so that
L(x) = X = { y | y < x }

Notice that
x =< y <-> L(x) sub L(y)

We can extend the image of L in R (the Dedekind cuts in R)
with the sets R and {}.
(Remember, L(R) doesn't have reals, it has sets of reals.)
Let's call L(R) U { R, {} } = L(R)+

Notice that
all x in L(R)+, x sub R
and
all x in L(R)+, {} sub x

If we extend R to R+ with two constants +inf, -inf
We could extend our infection L(x) to L'(x) so that
L': R+ -> L(R)+
with
L(+inf) = R
and
L(-inf) = {}

Then we would still have
x < y <-> L'(x) sub L'(y)
only now with two more elements such that
all x in R+, x =< +inf
and
all x in R+, -inf =< x

If I'm wrong, I hope someone corrects me, but that
looks like the reals extended with positive and negative
infinity.
Dan Christensen
2017-07-22 04:00:15 UTC
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Post by Jim Burns
On Friday, July 21, 2017 at 9:31:47 PM UTC-4,
On Friday, July 21, 2017 at 6:04:12 PM UTC-7,
Post by Dan Christensen
Given the real numbers, what axioms of ZFC would you
use to extend them to include an infinite number?
Axiom of Union
The union of what set(s)?
Using one-sided Dedekind cuts,
the union of the set of real numbers is just
the set of rational numbers.
Yes, the set of rational numbers is infinite,
That's a different kind of infinity, cardinality.
but it would be a stretch to call it the extension
of the reals to "infinity."
Recall that you gave the real numbers.
For each real number x (of the unextended reals),
there is exactly one set of real numbers X = {y| y < x}
the set of reals y less than x.
Define an injection L: R -> P(R) so that
L(x) = X = { y | y < x }
Notice that
x =< y <-> L(x) sub L(y)
We can extend the image of L in R (the Dedekind cuts in R)
with the sets R and {}.
(Remember, L(R) doesn't have reals, it has sets of reals.)
Let's call L(R) U { R, {} } = L(R)+
How do you go from this set to the notion of some kind of "infinite number?" And that it might somehow be equal to 1/0?

Dan
Post by Jim Burns
Notice that
all x in L(R)+, x sub R
and
all x in L(R)+, {} sub x
If we extend R to R+ with two constants +inf, -inf
We could extend our infection L(x) to L'(x) so that
L': R+ -> L(R)+
with
L(+inf) = R
and
L(-inf) = {}
Then we would still have
x < y <-> L'(x) sub L'(y)
only now with two more elements such that
all x in R+, x =< +inf
and
all x in R+, -inf =< x
If I'm wrong, I hope someone corrects me, but that
looks like the reals extended with positive and negative
infinity.
Jim Burns
2017-07-22 16:09:24 UTC
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On Friday, July 21, 2017 at 10:13:35 PM UTC-4,
Post by Jim Burns
On Friday, July 21, 2017 at 9:31:47 PM UTC-4,
On Friday, July 21, 2017 at 6:04:12 PM UTC-7,
Post by Dan Christensen
Given the real numbers, what axioms of ZFC would you
use to extend them to include an infinite number?
Axiom of Union
The union of what set(s)?
Using one-sided Dedekind cuts,
the union of the set of real numbers is just
the set of rational numbers.
Yes, the set of rational numbers is infinite,
That's a different kind of infinity, cardinality.
but it would be a stretch to call it the extension
of the reals to "infinity."
Recall that you gave the real numbers.
For each real number x (of the unextended reals),
there is exactly one set of real numbers X = {y| y < x}
the set of reals y less than x.
Define an injection L: R -> P(R) so that
L(x) = X = { y | y < x }
Notice that
x =< y <-> L(x) sub L(y)
We can extend the image of L in R (the Dedekind cuts
in R) with the sets R and {}.
(Remember, L(R) doesn't have reals,
it has sets of reals.)
Let's call L(R) U { R, {} } = L(R)+
How do you go from this set to the notion of
some kind of "infinite number?"
And that it might somehow be equal to 1/0?
I'm going to assume that you've read all the way to
the end of my post. An important of the answer to
your question is "Those things I said next".

I'm assuming that we already have the usual set of real
numbers with their usual operations, and that we have
ZFC too.

What I'm doing is extending those usual real numbers with
a positive infinite number _larger than all_ usual, finite
real numbers and a negative infinite number _less than all_
usual, finite real numbers. It's certainly not the only way
to extend the reals to infinite numbers, but it's one way.

I'm not sure if you are concerned that the infinite
numbers I named +inf and -inf exist. Supposing you
are concerned, consider that the sets R and {}
that correspond to them do exist (by assumption, given
the reals and ZFC) and there is a bijection between R+ and
L(R)+ . Anything non-self-contradictory that we can say
about the elements of L(R)+ we should be able to translate
(using L(x)) into a statement about the elements of R+ .
As far as +inf and -inf are concerned, they exist because
(i) I said so, and (ii) they do not create contradictions
(no more than the reals or ZFC do). These are the same
reasons we say 0, 1, 2, ... exist.

It's not enough just to say that +inf and -inf exist.
Do they play well with others? How do the extended versions
of <, +, -, *, / work?

One thing I assumed is that we want the _usual_ part of
the extended operations to look like the _unextended_
operations. That bypasses deep discussions of whether
the finite, extended reals are "really" the reals. This
would be my great preference, to avoid that discussion.

I already showed how to extend =< by placing +inf at
the upper end of the extended reals and -inf at the
lower end of the extended reals.

If that was all I needed to do, there would have been
little point to introducing Dedekind cuts. But we can
define the extended +, -, *, / by looking at the
operation on finite unextended reals in those cuts.
This is what's done in the usual construction of the
real numbers from Dedekind cuts of the rationals.

By the way, _I won't know_ until I look whether this will
give us 1/0. I'm taking "larger (smaller) than all the
finite numbers" to be the defining characteristic of
+inf (-inf) and then seeing what follows from that.
(This is not the only way to do things.)

Notation:
[x] = { y e R | y < x }

Define an "addition" operation [+] on L(R)+
[x] [+] [y] = [z] <->
[z] = { w e R | (Eu e [x])(Ev e [y]( u + v = w ) }

where the '+' inside the definition of [z] is the usual
addition on the reals.

From this definition, we can see that, for all
_finite_ x and y
[x] [+] [y] = [x + y]

which is precisely what we want.
We also have, from the same definition,
[x] [+] [+inf]
= [x] [+] R
= R
= [+inf]
Therefore, it seems reasonable to define, for R+, x finite,
x + +inf = +inf
in order to preserve, for infinite numbers,
[x] [+] [y] = [x + y]

By similar arguments,
x + -inf = -inf
+inf + -inf = -inf
(This last one surprised me.)

We can try extending the definitions for
neg([x]) = [-x]
inv([x]) = [1/x]
[x][*][y] = [x*y]
as we extended
[x][+][y] = [x + y]
It might or might not give us something for 1/0.
Post by Jim Burns
Notice that
all x in L(R)+, x sub R
and
all x in L(R)+, {} sub x
If we extend R to R+ with two constants +inf, -inf
We could extend our infection L(x) to L'(x) so that
L': R+ -> L(R)+
with
L(+inf) = R
and
L(-inf) = {}
Then we would still have
x < y <-> L'(x) sub L'(y)
only now with two more elements such that
all x in R+, x =< +inf
and
all x in R+, -inf =< x
If I'm wrong, I hope someone corrects me, but that
looks like the reals extended with positive and negative
infinity.
Dan Christensen
2017-07-23 18:58:45 UTC
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Post by Jim Burns
On Friday, July 21, 2017 at 10:13:35 PM UTC-4,
Post by Jim Burns
On Friday, July 21, 2017 at 9:31:47 PM UTC-4,
On Friday, July 21, 2017 at 6:04:12 PM UTC-7,
Post by Dan Christensen
Given the real numbers, what axioms of ZFC would you
use to extend them to include an infinite number?
Axiom of Union
The union of what set(s)?
Using one-sided Dedekind cuts,
the union of the set of real numbers is just
the set of rational numbers.
Yes, the set of rational numbers is infinite,
That's a different kind of infinity, cardinality.
but it would be a stretch to call it the extension
of the reals to "infinity."
Recall that you gave the real numbers.
For each real number x (of the unextended reals),
there is exactly one set of real numbers X = {y| y < x}
the set of reals y less than x.
Define an injection L: R -> P(R) so that
L(x) = X = { y | y < x }
Notice that
x =< y <-> L(x) sub L(y)
We can extend the image of L in R (the Dedekind cuts
in R) with the sets R and {}.
(Remember, L(R) doesn't have reals,
it has sets of reals.)
Let's call L(R) U { R, {} } = L(R)+
How do you go from this set to the notion of
some kind of "infinite number?"
And that it might somehow be equal to 1/0?
I'm going to assume that you've read all the way to
the end of my post. An important of the answer to
your question is "Those things I said next".
I'm assuming that we already have the usual set of real
numbers with their usual operations, and that we have
ZFC too.
What I'm doing is extending those usual real numbers with
a positive infinite number _larger than all_ usual, finite
real numbers and a negative infinite number _less than all_
usual, finite real numbers. It's certainly not the only way
to extend the reals to infinite numbers, but it's one way.
I'm not sure if you are concerned that the infinite
numbers I named +inf and -inf exist.
They can be constructed using ZFC. OK.
Post by Jim Burns
Supposing you
are concerned, consider that the sets R and {}
that correspond to them do exist (by assumption, given
the reals and ZFC) and there is a bijection between R+ and
L(R)+ . Anything non-self-contradictory that we can say
about the elements of L(R)+ we should be able to translate
(using L(x)) into a statement about the elements of R+ .
As far as +inf and -inf are concerned, they exist because
(i) I said so,
AFAICT you can actually prove their existence using ZFC.
Post by Jim Burns
and (ii) they do not create contradictions
(no more than the reals or ZFC do).
If they did create contradictions, ZFC would be inconsistent.
Post by Jim Burns
These are the same
reasons we say 0, 1, 2, ... exist.
It's not enough just to say that +inf and -inf exist.
Do they play well with others? How do the extended versions
of <, +, -, *, / work?
You have to construct (i.e. prove the existence of) new versions of each operators. Probably not trivial.
Post by Jim Burns
One thing I assumed is that we want the _usual_ part of
the extended operations to look like the _unextended_
operations. That bypasses deep discussions of whether
the finite, extended reals are "really" the reals. This
would be my great preference, to avoid that discussion.
You would have to prove that the new versions of these new operators behave as required (which has yet to be defined).
Post by Jim Burns
I already showed how to extend =< by placing +inf at
the upper end of the extended reals and -inf at the
lower end of the extended reals.
If that was all I needed to do, there would have been
little point to introducing Dedekind cuts. But we can
define the extended +, -, *, / by looking at the
operation on finite unextended reals in those cuts.
This is what's done in the usual construction of the
real numbers from Dedekind cuts of the rationals.
By the way, _I won't know_ until I look whether this will
give us 1/0.
It all has to fit together. Division by zero is what started this whole discussion.
Post by Jim Burns
I'm taking "larger (smaller) than all the
finite numbers" to be the defining characteristic of
+inf (-inf) and then seeing what follows from that.
(This is not the only way to do things.)
[x] = { y e R | y < x }
Define an "addition" operation [+] on L(R)+
[x] [+] [y] = [z] <->
[z] = { w e R | (Eu e [x])(Ev e [y]( u + v = w ) }
This may be a requirement for your construction of the new addition function.
Post by Jim Burns
where the '+' inside the definition of [z] is the usual
addition on the reals.
From this definition, we can see that, for all
_finite_ x and y
[x] [+] [y] = [x + y]
which is precisely what we want.
We also have, from the same definition,
[x] [+] [+inf]
= [x] [+] R
= R
= [+inf]
Therefore, it seems reasonable to define, for R+, x finite,
x + +inf = +inf
in order to preserve, for infinite numbers,
[x] [+] [y] = [x + y]
By similar arguments,
x + -inf = -inf
+inf + -inf = -inf
(This last one surprised me.)
We can try extending the definitions for
neg([x]) = [-x]
inv([x]) = [1/x]
[x][*][y] = [x*y]
as we extended
[x][+][y] = [x + y]
It might or might not give us something for 1/0.
Aye, there's the rub!!!
Post by Jim Burns
Post by Jim Burns
Notice that
all x in L(R)+, x sub R
and
all x in L(R)+, {} sub x
If we extend R to R+ with two constants +inf, -inf
We could extend our infection L(x) to L'(x) so that
L': R+ -> L(R)+
with
L(+inf) = R
and
L(-inf) = {}
Then we would still have
x < y <-> L'(x) sub L'(y)
only now with two more elements such that
all x in R+, x =< +inf
and
all x in R+, -inf =< x
If I'm wrong, I hope someone corrects me, but that
looks like the reals extended with positive and negative
infinity.
FredJeffries
2017-07-22 04:59:41 UTC
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Post by Dan Christensen
Using one-sided Dedekind cuts, the union of the set of real numbers is just the set of rational numbers.
Really? (pun intended)
FredJeffries
2017-07-22 14:03:21 UTC
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Post by Dan Christensen
Using one-sided Dedekind cuts, the union of the set of real numbers is just the set of rational numbers.
Excellent support for the thesis that, what ever the real numbers ARE, they are not "one-sided Dedekind cuts".

isites.harvard.edu/fs/docs/icb.topic1240846.files/Benacerraf.pdf
FredJeffries
2017-07-22 04:57:02 UTC
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Post by Dan Christensen
Given the real numbers, what axioms of ZFC would you use to extend them to include an infinite number?
The projective line is (often) defined to be the set of lines through the origin the Cartesian plane R^2. The "finite" points are identified with the slopes of the non-vertical lines and the point at infinity with the line x=0.
Peter Percival
2017-07-22 16:52:19 UTC
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Post by Dan Christensen
Given the real numbers, what axioms of ZFC would you use to extend them to include an infinite number?
An interesting(*) thing to do is to broaden the notion of Dedekind cut
as in [1]. One then gets the real numbers, but also (with no further
effort, so to speak) a shockingly promiscuous collection of infinite and
infinitesimal numbers. The whole lot are called surreal numbers.

Also, in NBG (even though Mendelson says it stands for No Bloody Good,
it is as respectable as ZFC) one can define the largest ordered field
(all other ordered fields are (isomorphic to) subfields of it). At
least you can if you'll allow a field's domain to be a proper class. It
turns out to be the surreal numbers.

(* Interesting to me, not to you.)

[1] J.H. Conway, /On numbers and games/, A.K. Peters.
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
Peter Percival
2017-07-22 05:59:16 UTC
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Post by Dan Christensen
The natural numbers, integers, rationals, reals and complex numbers
can all apparently be constructed in ZFC theory. What about the
extension of the reals to "infinity?"
Are you asking if the one point compactification of the reals can be
defined in ZFC?
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
Dan Christensen
2017-07-22 13:24:29 UTC
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Post by Peter Percival
Post by Dan Christensen
The natural numbers, integers, rationals, reals and complex numbers
can all apparently be constructed in ZFC theory. What about the
extension of the reals to "infinity?"
Are you asking if the one point compactification of the reals can be
defined in ZFC?
To begin with, I am asking if something like an "infinite number" can be CONSTRUCTED (not simply "defined") using ONLY the ZFC axioms? Or will it require additional assumptions, e.g. about the existence of objects in addition to the empty set and the infinite set postulated in the ZFC axioms. Then, of course, we will need to construct the required addition and multiplication functions using only the ZFC axioms.

Dan

Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
FredJeffries
2017-07-22 13:54:52 UTC
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Post by Dan Christensen
To begin with, I am asking if something like an "infinite number" can be CONSTRUCTED (not simply "defined") using ONLY the ZFC axioms?
The projective line is (often) CONSTRUCTED to be the set of (CONSTRUCTED) lines through the origin the (CONSTRUCTED) Cartesian plane R^2. The "finite" points are identified with the slopes of the non-vertical lines and the point at infinity with the line x=0.
Peter Percival
2017-07-22 16:33:03 UTC
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On Saturday, July 22, 2017 at 1:59:31 AM UTC-4, Peter Percival
Post by Peter Percival
Post by Dan Christensen
The natural numbers, integers, rationals, reals and complex
numbers can all apparently be constructed in ZFC theory. What
about the extension of the reals to "infinity?"
Are you asking if the one point compactification of the reals can
be defined in ZFC?
To begin with, I am asking if something like an "infinite number" can
be CONSTRUCTED (not simply "defined") using ONLY the ZFC axioms? Or
First a confession: I have never been too sure what you mean by "construct".

But there are no axioms beyond the axioms of ZFC needed to do (let's
say) real analysis, and if we wish to bring the one point
compactification into our account of real analysis we will still not
need any more axioms.

(Yes, I am aware that there are questions one might ask about the real
numbers that aren't setted by the ZFC axioms. E.g. "is there a set of
real numbers the cardinality of which is greater than that of the
natural numbers but less than that of the whole set of real numbers?".
I don't think that detracts fro my uncontroversial claim that there are
no axioms beyond the axioms of ZFC needed to do real analysis.)
will it require additional assumptions, e.g. about the existence of
objects in addition to the empty set and the infinite set postulated
in the ZFC axioms.
I don't get that. ZFC grants us more sets than the empty set and the
("the"?) infinite set. For example this - {{}} is neither empty nor
infinite.
Then, of course, we will need to construct the
required addition and multiplication functions using only the ZFC
axioms.
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
Peter Percival
2017-07-22 16:59:47 UTC
Reply
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Post by Peter Percival
Post by Dan Christensen
Post by Peter Percival
Post by Dan Christensen
The natural numbers, integers, rationals, reals and complex
numbers can all apparently be constructed in ZFC theory. What
about the extension of the reals to "infinity?"
Are you asking if the one point compactification of the reals
can be defined in ZFC?
To begin with, I am asking if something like an "infinite number"
can be CONSTRUCTED (not simply "defined") using ONLY the ZFC
axioms? Or
First a confession: I have never been too sure what you mean by "construct".
But there are no axioms beyond the axioms of ZFC needed to do (let's
say) real analysis, and if we wish to bring the one point
compactification into our account of real analysis we will still not
need any more axioms.
(Yes, I am aware that there are questions one might ask about the
real numbers that aren't setted by the ZFC axioms. E.g. "is there a
aren't settled by
Post by Peter Percival
set of
real numbers the cardinality of which is greater than that of the
natural numbers but less than that of the whole set of real
numbers?". I don't think that detracts fro my uncontroversial claim
detracts from
Post by Peter Percival
that there are no axioms beyond the axioms of ZFC needed to do real
analysis.)
Post by Dan Christensen
will it require additional assumptions, e.g. about the existence
of objects in addition to the empty set and the infinite set
postulated in the ZFC axioms.
I don't get that. ZFC grants us more sets than the empty set and
the ("the"?) infinite set. For example this - {{}} is neither empty
nor infinite.
Post by Dan Christensen
Then, of course, we will need to construct the required addition
and multiplication functions using only the ZFC axioms.
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
Dan Christensen
2017-07-23 19:36:11 UTC
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Post by Peter Percival
Post by Dan Christensen
Post by Peter Percival
Post by Dan Christensen
The natural numbers, integers, rationals, reals and complex
numbers can all apparently be constructed in ZFC theory. What
about the extension of the reals to "infinity?"
Are you asking if the one point compactification of the reals can
be defined in ZFC?
To begin with, I am asking if something like an "infinite number" can
be CONSTRUCTED (not simply "defined") using ONLY the ZFC axioms? Or
First a confession: I have never been too sure what you mean by "construct".
Only 2 sets are postulated in the ZFC axioms: The empty set and the set postulated to exist by the Axiom of Infinity. Every other set in ZFC theory must be "constructed" from only these 2 sets using only the axioms of ZFC. The set of natural numbers, for example, can be "constructed" by using the Axiom of Specification (or Subsets) to extract it from the set postulated in AOI.
Post by Peter Percival
But there are no axioms beyond the axioms of ZFC needed to do (let's
say) real analysis, and if we wish to bring the one point
compactification into our account of real analysis we will still not
need any more axioms.
I am not yet convinced that a the required set of extended reals and the required operators can be "constructed" using only the Axioms of ZFC in such a way as to model division by zero in any meaningful way. Yes, this is somewhat ambiguous and perhaps gives me too much wiggle room, but I don't know how else to say it.
Post by Peter Percival
(Yes, I am aware that there are questions one might ask about the real
numbers that aren't setted by the ZFC axioms. E.g. "is there a set of
real numbers the cardinality of which is greater than that of the
natural numbers but less than that of the whole set of real numbers?".
I don't think that detracts fro my uncontroversial claim that there are
no axioms beyond the axioms of ZFC needed to do real analysis.)
Post by Dan Christensen
will it require additional assumptions, e.g. about the existence of
objects in addition to the empty set and the infinite set postulated
in the ZFC axioms.
I don't get that. ZFC grants us more sets than the empty set and the
("the"?) infinite set. For example this - {{}} is neither empty nor
infinite.
The existence of other sets like this can be inferred only by using the ZFC axioms. I'm just wondering wondering if the empty and infinite sets postulated to exist in these axioms are a sufficient starting point for attaching any meaning to division by zero. I'm guessing it isn't.
Post by Peter Percival
Post by Dan Christensen
Then, of course, we will need to construct the
required addition and multiplication functions using only the ZFC
axioms.
--
Dan

Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
Peter Percival
2017-07-23 19:55:14 UTC
Reply
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Post by Dan Christensen
Post by Peter Percival
Post by Dan Christensen
Post by Peter Percival
Post by Dan Christensen
The natural numbers, integers, rationals, reals and complex
numbers can all apparently be constructed in ZFC theory.
What about the extension of the reals to "infinity?"
Are you asking if the one point compactification of the reals
can be defined in ZFC?
To begin with, I am asking if something like an "infinite number"
can be CONSTRUCTED (not simply "defined") using ONLY the ZFC
axioms? Or
First a confession: I have never been too sure what you mean by "construct".
Only 2 sets are postulated in the ZFC axioms: The empty set and the
set postulated to exist by the Axiom of Infinity. Every other set in
ZFC theory must be "constructed" from only these 2 sets using only
the axioms of ZFC. The set of natural numbers, for example, can be
"constructed" by using the Axiom of Specification (or Subsets) to
extract it from the set postulated in AOI.
I'm puzzled by the above, but I've now read a post from you elsewhere in
this thread, you write

You have to construct (i.e. prove the existence of) [...]

"Prove the existence of" I understand. Usually one wants to prove
uniqueness as well. Ok.
Post by Dan Christensen
Post by Peter Percival
But there are no axioms beyond the axioms of ZFC needed to do
(let's say) real analysis, and if we wish to bring the one point
compactification into our account of real analysis we will still
not need any more axioms.
I am not yet convinced that a the required set of extended reals and
the required operators can be "constructed" using only the Axioms of
ZFC in such a way as to model division by zero in any meaningful way.
Yes, this is somewhat ambiguous and perhaps gives me too much wiggle
room, but I don't know how else to say it.
Post by Peter Percival
(Yes, I am aware that there are questions one might ask about the
real numbers that aren't setted by the ZFC axioms. E.g. "is there
a set of real numbers the cardinality of which is greater than that
of the natural numbers but less than that of the whole set of real
numbers?". I don't think that detracts fro my uncontroversial claim
that there are no axioms beyond the axioms of ZFC needed to do real
analysis.)
Post by Dan Christensen
will it require additional assumptions, e.g. about the existence
of objects in addition to the empty set and the infinite set
postulated in the ZFC axioms.
I don't get that. ZFC grants us more sets than the empty set and
the ("the"?) infinite set. For example this - {{}} is neither
empty nor infinite.
The existence of other sets like this can be inferred only by using
the ZFC axioms. I'm just wondering wondering if the empty and
infinite sets postulated to exist in these axioms are a sufficient
starting point for attaching any meaning to division by zero. I'm
guessing it isn't.
So it's only your guess?
Post by Dan Christensen
Post by Peter Percival
Post by Dan Christensen
Then, of course, we will need to construct the required addition
and multiplication functions using only the ZFC axioms.
--
Dan
Download my DC Proof 2.0 software at http://www.dcproof.com Visit my
Math Blog at http://www.dcproof.wordpress.com
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
Peter Percival
2017-07-23 21:07:42 UTC
Reply
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Post by Peter Percival
Post by Dan Christensen
Post by Peter Percival
Post by Dan Christensen
Post by Peter Percival
Post by Dan Christensen
The natural numbers, integers, rationals, reals and complex
numbers can all apparently be constructed in ZFC theory.
What about the extension of the reals to "infinity?"
Are you asking if the one point compactification of the reals
can be defined in ZFC?
To begin with, I am asking if something like an "infinite number"
can be CONSTRUCTED (not simply "defined") using ONLY the ZFC
axioms? Or
First a confession: I have never been too sure what you mean by "construct".
Only 2 sets are postulated in the ZFC axioms: The empty set and the
set postulated to exist by the Axiom of Infinity. Every other set in
ZFC theory must be "constructed" from only these 2 sets using only
the axioms of ZFC. The set of natural numbers, for example, can be
"constructed" by using the Axiom of Specification (or Subsets) to
extract it from the set postulated in AOI.
I'm puzzled by the above, but I've now read a post from you elsewhere in
this thread, you write
You have to construct (i.e. prove the existence of) [...]
"Prove the existence of" I understand. Usually one wants to prove
uniqueness as well. Ok.
So now I get the meaning of your "construct", your question above
repeated here:

To begin with, I am asking if something like an "infinite number" can
be CONSTRUCTED (not simply "defined") using ONLY the ZFC axioms?

can be answered yes.
Post by Peter Percival
Post by Dan Christensen
Post by Peter Percival
But there are no axioms beyond the axioms of ZFC needed to do
(let's say) real analysis, and if we wish to bring the one point
compactification into our account of real analysis we will still
not need any more axioms.
I am not yet convinced that a the required set of extended reals and
the required operators can be "constructed" using only the Axioms of
ZFC in such a way as to model division by zero in any meaningful way.
Yes, this is somewhat ambiguous and perhaps gives me too much wiggle
room, but I don't know how else to say it.
Post by Peter Percival
(Yes, I am aware that there are questions one might ask about the
real numbers that aren't setted by the ZFC axioms. E.g. "is there
a set of real numbers the cardinality of which is greater than that
of the natural numbers but less than that of the whole set of real
numbers?". I don't think that detracts fro my uncontroversial claim
that there are no axioms beyond the axioms of ZFC needed to do real
analysis.)
Post by Dan Christensen
will it require additional assumptions, e.g. about the existence
of objects in addition to the empty set and the infinite set
postulated in the ZFC axioms.
I don't get that. ZFC grants us more sets than the empty set and
the ("the"?) infinite set. For example this - {{}} is neither
empty nor infinite.
The existence of other sets like this can be inferred only by using
the ZFC axioms. I'm just wondering wondering if the empty and
infinite sets postulated to exist in these axioms are a sufficient
starting point for attaching any meaning to division by zero. I'm
guessing it isn't.
So it's only your guess?
Post by Dan Christensen
Post by Peter Percival
Post by Dan Christensen
Then, of course, we will need to construct the required addition
and multiplication functions using only the ZFC axioms.
--
Dan
Download my DC Proof 2.0 software at http://www.dcproof.com Visit my
Math Blog at http://www.dcproof.wordpress.com
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
Peter Percival
2017-07-22 17:01:34 UTC
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[...] Then, of course, we will need to construct the required
addition andmultiplication functions using only the ZFC axioms.
I don't know why, but now I can't get "Buddy can you spare a dime" out
of my head.
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
Shobe, Martin
2017-07-22 21:46:37 UTC
Reply
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Post by Dan Christensen
Post by Peter Percival
Post by Dan Christensen
The natural numbers, integers, rationals, reals and complex numbers
can all apparently be constructed in ZFC theory. What about the
extension of the reals to "infinity?"ONS
Are you asking if the one point compactification of the reals can be
defined in ZFC?
To begin with, I am asking if something like an "infinite number" can be CONSTRUCTED (not simply "defined") using ONLY the ZFC axioms?
I'm not sure what sort of distinction you are trying to make between
CONSTRUCTED and "defined" here. The standard constructions of things
like the naturals, integers, rationals, reals, etc. are simply
definitions that happen to result in a set and functions with the
desired properties. That can certainly be done using "ONLY the ZFC
axioms". If you mean something else, as I suspect you are, you'll have
to give us your private definitions of the terms you are using before
your question can be adequately answered.
Post by Dan Christensen
Or will it require additional assumptions, e.g. about the existence of objects in addition to the empty set and the infinite set postulated in the ZFC axioms.
No. We just need the appropriate definitions.
Post by Dan Christensen
Then, of course, we will need to construct the required addition and multiplication functions using only the ZFC axioms.
Yes, those will need to be defined too.

Martin Shobe
Dan Christensen
2017-07-23 20:04:28 UTC
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Post by Shobe, Martin
Post by Dan Christensen
Post by Peter Percival
Post by Dan Christensen
The natural numbers, integers, rationals, reals and complex numbers
can all apparently be constructed in ZFC theory. What about the
extension of the reals to "infinity?"ONS
Are you asking if the one point compactification of the reals can be
defined in ZFC?
To begin with, I am asking if something like an "infinite number" can be CONSTRUCTED (not simply "defined") using ONLY the ZFC axioms?
I'm not sure what sort of distinction you are trying to make between
CONSTRUCTED and "defined" here. The standard constructions of things
like the naturals, integers, rationals, reals, etc. are simply
definitions that happen to result in a set and functions with the
desired properties.
Example: Given the set of natural numbers N, Peano's Axioms and the set of ordered triples NxNxN, we can CONSTRUCT (i.e. prove the existence of) the add function on N as a subset of the NxNxN by using the Specification (or Subset) Axiom):

For all a,b,c: [(a,b,c) in Add <=> (a,b,c) in NxNxN
& For all d: [d is a subset of NxNxN
& For all e: [e in N => (e,0,e) in d]
& For all e,f,g: [(e,f,g) in d => (e,S(f),S(g)) in d]
=> (a,b,c) in d]]

We must then PROVE (not simply define) that Add (as constructed above) is a function and that:

1. For a: [a in N => Add(a,0)=a]

2. For all a, b, c in N: [Add(a,b)=c) => Add(a,S(b))=S(c)]

Dan

Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
FredJeffries
2017-07-21 12:27:08 UTC
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Post by Dan Christensen
You seem to be suggesting that these other "systems" are, in some sense, equivalent to that which is widely used, and that they are somehow equally useful in applications. That is pure nonsense, of course.
What happens when you have a spanking brand new computer and some doofus comes along and enters: PRINT 1 / 0

Does the CPU instantly fry? Or does the entire universe come to an end?
Peter Percival
2017-07-22 22:29:47 UTC
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Post by Doug
I remember back in my school days when this came up we were told that
the simple answer is it can't be done, but that mathematicians have
got it figured out, it's just too complicated for normal humans.
What age were you? That one cannot divide by zero in (say) the rational
or real numbers is easy enough to explain, I would have thought.
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
Doug
2017-07-20 16:23:43 UTC
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Post by Julio Di Egidio
As some have tried to explain, division by zero is
defined in some systems,
not defined in others. There is no mystery about it,
just insane bullshit
by the resident spammers.
HTH,
Julio
Thanks... I have a passing interest in the subject, was looking for input from people who know a lot more than me. Of course this subject can easily go off the deep end and probably starts a lot of silly and opinionated arguments.
Jonathan Cender
2017-07-21 02:44:47 UTC
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Hi Doug. I have done considerable work in this area and have a novel approach - redefining zero so that it is not a real number. Instead of it being real, zero becomes a number in another set of numbers where division of reals by zero makes sense in a way somewhat similar to the square root of -1 making sense in the imaginary numbers. Links to my work follow.

First let me say that I agree with those who say there is no "official" answer in the sense you mean. It is defined in some ways for certain purposes as with the Riemann Sphere. The Wolfram Alpha math/search engine (http://www.wolframalpha.com/) uses the Riemann approach in answering the entry "1/0" for example. The link to the Cantrell post provided to you in an earlier post gives a different approach for a different purpose while the many who work with Meadows (totalized fields) and Carlstrom's Wheels give yet other approaches.
All of these approaches to division by 0, and more, are covered in a general way in a chapter in Patrick Suppes's Introduction to Logic by the way. Except for Cantrell, references to these approaches and more are given in my paper.

For more discussion on Quora, including a link to my paper at the end, go here https://www.quora.com/Why-cant-we-come-up-with-a-corresponding-symbol-for-the-question-What-is-1-divided-by-0/answer/Jonathan-Cender

and a direct link to my paper, Replacing 0 - A NonEuclidean Arithmetic, https://drive.google.com/file/d/0B46E-e1cnYi6TDRJV3U5VUUzUVk/view?usp=sharing
7777777
2017-07-21 20:09:20 UTC
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Post by Jonathan Cender
Hi Doug. I have done considerable work in this area and have a novel approach - redefining zero so that it is not a real number.
what do you think about 0.000...001 = 0 ?
Peter Percival
2017-07-21 20:19:03 UTC
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perjantai 21. heinäkuuta 2017 13.30.09 UTC+3 Jonathan Cender
Post by Jonathan Cender
Hi Doug. I have done considerable work in this area and have a
novel approach - redefining zero so that it is not a real number.
what do you think about 0.000...001 = 0 ?
How many digits 0 do those three dots represent?
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
7777777
2017-07-24 18:46:55 UTC
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Post by Peter Percival
Post by 7777777
what do you think about 0.000...001 = 0 ?
How many digits 0 do those three dots represent?
1/10 = 0.1 -> number of zeros 1
1/10^2 = 0.01 -> number of zeros 2
1/10^3 = 0.001 -> number of zeros 3
.
.
.
1/10^Z = 0.000...001 -> number of zeros Z
Peter Percival
2017-07-25 10:25:37 UTC
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Post by 7777777
Post by Peter Percival
Post by 7777777
what do you think about 0.000...001 = 0 ?
How many digits 0 do those three dots represent?
1/10 = 0.1 -> number of zeros 1
1/10^2 = 0.01 -> number of zeros 2
1/10^3 = 0.001 -> number of zeros 3
.
.
.
1/10^Z = 0.000...001 -> number of zeros Z
If I understand you (that's a big "if"), 0.000...001 doesn't = 0.
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
Jonathan Cender
2017-07-21 19:18:33 UTC
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Try 1/0 at http://www.wolframalpha.com/
Wolfram Alpha gives an answer using the math of the extended complex plane (Riemann Sphere).
Simon Roberts
2017-07-21 23:13:50 UTC
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Post by Doug
Is this one of those unsolved mysteries of Mathematics, or is it just too high level for the average person? I tried to search but didn't find anything that looked like an "official" or commonly accepted answer.
Thanks!
I cannot answer your question(s).
[BTW. Must I choose between only two possible answers? "tongue 'n cheek"].

This link is to a text file that may lead you to a semblance of "one" particular answer. Not official, by any means I know today, and a little sketchy.

"I hope this helps."

https://www.dropbox.com/s/g9ixi6oir0a2g59/Full.txt?dl=0

Simon Roberts
Jonathan Cender
2017-07-21 22:06:43 UTC
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I think that 0.000...001 > 0

And not = 0
perjantai 21. heinäkuuta 2017 13.30.09 UTC+3 Jonathan
Post by Jonathan Cender
Hi Doug. I have done considerable work in this area
and have a novel approach - redefining zero so that
it is not a real number.
what do you think about 0.000...001 = 0 ?
7777777
2017-07-25 06:12:38 UTC
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Post by Jonathan Cender
I think that 0.000...001 > 0
And not = 0
alright, but then think about those who vehemently deny the existence
of nonzero infinitesimals. What about them?
Jim Burns
2017-07-25 13:17:43 UTC
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lauantai 22. heinäkuuta 2017 4.00.11 UTC+3
Post by Jonathan Cender
I think that 0.000...001 > 0
And not = 0
alright, but then think about those who vehemently
deny the existence of nonzero infinitesimals.
What about them?
Many of those who deny the existence of nonzero
infinitesimals are speaking of non-existence
_in the real numbers_ , in the complete ordered field.
There, it's not a matter of opinion that there aren't
any nonzero infinitesimals. There aren't.

The vehemence involved in that statement is irrelevant,
like the font that the statement is typed in, or the
color of the paper it's printed on.

Some people who claim that nonzero infinitesimals exist
are talking about existence in something _other than_
the real numbers. Many of _these_ people would very likely
also vehemently deny that there are nonzero infinitesimals
_in the real numbers_ .

What about them? They're correct.

Some people who claim that nonzero infinitesimals exist
are either making that claim about the real numbers or
don't understand the distinction between formal systems.
(Aren't _all_ numbers really _real_ numbers when you
get down to it? Answer: No, they aren't.)

What about them? They're wrong or not-even-wrong.

They are fairly unusual in having an interest in this
area, which is an excellent beginning to learning about
it. But they need to continue on and actually learn
about infinitesimals and so on, instead of just
assuming that whatever popped into their minds unbidden
is how things are.
7777777
2017-07-25 17:28:54 UTC
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Post by Jim Burns
Many of those who deny the existence of nonzero
infinitesimals are speaking of non-existence
_in the real numbers_ , in the complete ordered field.
There, it's not a matter of opinion that there aren't
any nonzero infinitesimals. There aren't.
The vehemence involved in that statement is irrelevant,
like the font that the statement is typed in, or the
color of the paper it's printed on.
Some people who claim that nonzero infinitesimals exist
are talking about existence in something _other than_
the real numbers. Many of _these_ people would very likely
also vehemently deny that there are nonzero infinitesimals
_in the real numbers_ .
What about them? They're correct.
not necessarily:
their claim is based on real numbers being unambiguous, while the truth is
that they are not.

Also, consider adding these "your" nonzero infinitesimals. How many of them do
you need to add together in order for their sum to appear in
real numbers? Don't you in this case have infinitesimals in real numbers?
Jim Burns
2017-07-25 17:41:51 UTC
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tiistai 25. heinäkuuta 2017 16.17.52 UTC+3
Post by Jim Burns
Many of those who deny the existence of nonzero
infinitesimals are speaking of non-existence
_in the real numbers_ , in the complete ordered field.
There, it's not a matter of opinion that there aren't
any nonzero infinitesimals. There aren't.
The vehemence involved in that statement is irrelevant,
like the font that the statement is typed in, or the
color of the paper it's printed on.
Some people who claim that nonzero infinitesimals exist
are talking about existence in something _other than_
the real numbers. Many of _these_ people would very likely
also vehemently deny that there are nonzero infinitesimals
_in the real numbers_ .
What about them? They're correct.
their claim is based on real numbers being unambiguous,
Their claim is based on the real numbers unambiguously
having certain properties, their axioms.
while the truth is that they are not.
_What the real numbers are_ is ambiguous.
_Whether the real numbers have nonzero infinitesimals_
is not ambiguous.
Also, consider adding these "your" nonzero infinitesimals.
Okay.
So, we're not talking about real numbers now.
How many of them do you need to add together in order
for their sum to appear in real numbers? Don't you in
this case have infinitesimals in real numbers?
No.
7777777
2017-07-25 18:25:09 UTC
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Post by Jim Burns
Post by 7777777
Also, consider adding these "your" nonzero infinitesimals.
Okay.
So, we're not talking about real numbers now.
Post by 7777777
How many of them do you need to add together in order
for their sum to appear in real numbers? Don't you in
this case have infinitesimals in real numbers?
No.
consider the sum
0.000...001+0.000...001+0.000...001+.....
how many of them do I need to add in order for their sum to equal to 1 ?
And don't, in this case, the number 1 consist of infinitesimals?
How should this situation differ from there not being infinitesimals in
real numbers?
Peter Percival
2017-07-25 20:59:37 UTC
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consider the sum 0.000...001+0.000...001+0.000...001+..... how many
of them do I need to add in order for their sum to equal to 1 ? And
Are you asking how many infinitesimals need to be added together to get
1? That seems like a reasonable question, but writing 0.000...001 for
an infinitesimal isn't reasonable notation.
don't, in this case, the number 1 consist of infinitesimals? How
should this situation differ from there not being infinitesimals in
real numbers?
I'm not convinced. Writing "i" for either square root of -1:

(1 + i) + (1 - i) = 2.

Does 2 being real mean that i is in the real numbers?
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
Julio Di Egidio
2017-07-26 01:36:47 UTC
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Post by Peter Percival
I'm not convinced.
And you too are just a spammer, systematically stirring nonsense and
nutcases in both sci.math and sci.logic.

Julio
Jim Burns
2017-07-25 21:27:43 UTC
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tiistai 25. heinäkuuta 2017 20.41.58 UTC+3
Post by Jim Burns
Post by 7777777
Also, consider adding these "your" nonzero
infinitesimals.
Okay.
So, we're not talking about real numbers now.
Post by 7777777
How many of them do you need to add together in order
for their sum to appear in real numbers? Don't you in
this case have infinitesimals in real numbers?
No.
consider the sum
0.000...001+0.000...001+0.000...001+.....
how many of them do I need to add in order for
their sum to equal to 1 ?
You told Peter Percival earlier
<7>
1/10^Z = 0.000...001 -> number of zeros Z
</7>
[Date: Mon, 24 Jul 2017 11:46:55 -0700 (PDT)]
although I think you meant "number of zeros Z-1"

It would take 10^Z of them added together to get 1.
And don't, in this case, the number 1 consist of
infinitesimals?
No.

Z is some positive integer, possibly a very large
positive integer.
(Although, for reasonable definitions of "very large
positive integer", one can prove that there are no
very large positive integers.)

_For all Z_ 1/10^Z is not an infinitesimal.
How should this situation differ from there not
being infinitesimals in real numbers?
I'm not sure that we are communicating.

I would say that there are _still_ no nonzero infinitesimals
in real numbers. So, I would say that this is the situation
that you describe. So, I would say that it doesn't differ
from itself.

But maybe you meant to ask something else.
Peter Percival
2017-07-25 14:44:50 UTC
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Post by 7777777
Post by Jonathan Cender
I think that 0.000...001 > 0
And not = 0
alright, but then think about those who vehemently deny the existence
of nonzero infinitesimals. What about them?
I asked not long ago

'How many digits 0 do those three dots represent?'

And you replied

'1/10 = 0.1 -> number of zeros 1
1/10^2 = 0.01 -> number of zeros 2
1/10^3 = 0.001 -> number of zeros 3
.
.
.
1/10^Z = 0.000...001 -> number of zeros Z'

Which I took (silly me) to mean that those three dots represented an
unspecified, but finite, number of 0's. If that's so then Jonathan
Cender's post is correct. And the existence of nonzero infinitesimals
has no relevance to the matter.
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
Jonathan Cender
2017-07-21 22:27:37 UTC
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Try 1/0 at http://www.wolframalpha.com/
Wolfram Alpha gives an answer using the math of the extended complex plane (Riemann Sphere).
Post by FredJeffries
What happens when you have a spanking brand new
PRINT 1 / 0
Does the CPU instantly fry? Or does the entire
universe come to an end?
Doug
2017-07-22 00:19:12 UTC
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Post by Jonathan Cender
Hi Doug. I have done considerable work in this area
and have a novel approach - redefining zero so that
it is not a real number. Instead of it being real,
zero becomes a number in another set of numbers where
division of reals by zero makes sense in a way
somewhat similar to the square root of -1 making
sense in the imaginary numbers. Links to my work
follow.
First let me say that I agree with those who say
there is no "official" answer in the sense you mean.
It is defined in some ways for certain purposes as
with the Riemann Sphere. The Wolfram Alpha
math/search engine (http://www.wolframalpha.com/)
uses the Riemann approach in answering the entry
"1/0" for example. The link to the Cantrell post
provided to you in an earlier post gives a different
approach for a different purpose while the many who
work with Meadows (totalized fields) and Carlstrom's
Wheels give yet other approaches.
All of these approaches to division by 0, and more,
are covered in a general way in a chapter in Patrick
Suppes's Introduction to Logic by the way. Except for
Cantrell, references to these approaches and more are
given in my paper.
For more discussion on Quora, including a link to my
paper at the end, go here
https://www.quora.com/Why-cant-we-come-up-with-a-corre
sponding-symbol-for-the-question-What-is-1-divided-by-
0/answer/Jonathan-Cender
and a direct link to my paper, Replacing 0 - A
NonEuclidean Arithmetic,
https://drive.google.com/file/d/0B46E-e1cnYi6TDRJV3U5V
UUzUVk/view?usp=sharing
Thanks! I will surely check out your stuff!
Doug
2017-07-22 00:20:57 UTC
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Post by Jonathan Cender
Try 1/0 at http://www.wolframalpha.com/
Wolfram Alpha gives an answer using the math of the
extended complex plane (Riemann Sphere).
Haha yes thanks for the link... infinity is the obvious answer, but everyone shoots it down for the obvious reasons o_O
Simon Roberts
2017-07-22 02:11:22 UTC
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Post by Doug
Post by Jonathan Cender
Try 1/0 at http://www.wolframalpha.com/
Wolfram Alpha gives an answer using the math of the
extended complex plane (Riemann Sphere).
Haha yes thanks for the link... infinity is the obvious answer, but everyone shoots it down for the obvious reasons o_O
call it what you want.
Doug
2017-07-22 00:31:58 UTC
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Post by Simon Roberts
I cannot answer your question(s).
[BTW. Must I choose between only two possible
answers? "tongue 'n cheek"].
This link is to a text file that may lead you to a
semblance of "one" particular answer. Not official,
by any means I know today, and a little sketchy.
"I hope this helps."
https://www.dropbox.com/s/g9ixi6oir0a2g59/Full.txt?dl=
0
Simon Roberts
Thanks. Unfortunately I am not familiar with the notation. Maybe if I dig into this stuff enough I can return to that and understand it, but right now I'm a first grader I guess.

On your joke, I guess it's a leading question, but I am at least smart enough to know how far ranging (infinite?) the replies can become, so I did try to "guide" it a little :D

I also threw in the caveat of "commonly accepted" and put "official" in quotes, didn't help much.
Jonathan Cender
2017-07-22 02:17:34 UTC
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On Friday, July 21, 2017 at 8:35:16 PM UTC-4, Shobe,
Given the real numbers, what axioms of ZFC would you
use to extend them to include an infinite number?
Existing ZFC axioms are used. Crucially, a new set, the "absent set," is substituted at relevant places for the empty set. There is a section in my paper devoted to this new set.

Apart from ZFC, the definition for the new infinite numbers involves a notation useful for n-real dimensional space from Roger Penrose and John A Wheeler. The notation, Wallis's familiar infinity symbol, represents "a one line array of the real numbers."

So in ZFC, axioms, not necessarily new ones, would need to be understood as referring to the reals as an array number. So axioms that would be the same or similar to those used as a basis for transfinites, but applying to "array" numbers. In other words a new interpretation might be enough. I think so.

https://drive.google.com/file/d/0B46E-e1cnYi6TDRJV3U5VUUzUVk/view?usp=sharing
Doug
2017-07-23 17:08:50 UTC
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Post by Doug
Post by Doug
I remember back in my school days when this came up
we were told that
Post by Doug
the simple answer is it can't be done, but that
mathematicians have
Post by Doug
got it figured out, it's just too complicated for
normal humans.
What age were you? That one cannot divide by zero in
(say) the rational
or real numbers is easy enough to explain, I would
have thought.
Anywhere before the point that a person must decide on a career path, especially one that is academically intensive.
Jonathan Cender
2017-07-24 20:17:54 UTC
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Doug,

Please note. Wolfram does not give "infinity" as the answer. Yes, the infinity symbol is used. However, reading more closely on that page shows that the answer is "complex infinity." The definition used is from the extended complex plane (Riemann Sphere).
Post by Doug
Post by Jonathan Cender
Try 1/0 at http://www.wolframalpha.com/
Wolfram Alpha gives an answer using the math of the
extended complex plane (Riemann Sphere).
Haha yes thanks for the link... infinity is the
obvious answer, but everyone shoots it down for the
obvious reasons o_O
Doug
2017-07-24 21:28:21 UTC
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Thanks, yes I did notice that, it is a slightly different symbol. I think the Riemann Sphere is a partial attempt at solving the problem, but really leaves the same fundamental question hanging.
Post by Jonathan Cender
Doug,
Please note. Wolfram does not give "infinity" as the
answer. Yes, the infinity symbol is used. However,
reading more closely on that page shows that the
answer is "complex infinity." The definition used is
from the extended complex plane (Riemann Sphere).
Post by Doug
Post by Jonathan Cender
Try 1/0 at http://www.wolframalpha.com/
Wolfram Alpha gives an answer using the math of
the
Post by Doug
Post by Jonathan Cender
extended complex plane (Riemann Sphere).
Haha yes thanks for the link... infinity is the
obvious answer, but everyone shoots it down for the
obvious reasons o_O
Bill
2017-07-24 23:06:26 UTC
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Post by Doug
Thanks, yes I did notice that, it is a slightly different symbol. I think the Riemann Sphere is a partial attempt at solving the problem, but really leaves the same fundamental question hanging.
Think of the graph of y=1/x, or y=1/z. There is no fundamental question
hanging--at least not for anyone who has done any sort of math
coursework, or trolls.
Doug
2017-07-25 22:39:59 UTC
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Post by Doug
Post by Doug
Thanks, yes I did notice that, it is a slightly
different symbol. I think the Riemann Sphere is a
partial attempt at solving the problem, but really
leaves the same fundamental question hanging.
Think of the graph of y=1/x, or y=1/z. There is no
fundamental question
hanging--at least not for anyone who has done any
sort of math
coursework, or trolls.
Once upon a time, I would have believed that :D
Bill
2017-07-25 23:25:06 UTC
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Post by Doug
Post by Doug
Post by Doug
Thanks, yes I did notice that, it is a slightly
different symbol. I think the Riemann Sphere is a
partial attempt at solving the problem, but really
leaves the same fundamental question hanging.
Think of the graph of y=1/x, or y=1/z. There is no
fundamental question
hanging--at least not for anyone who has done any
sort of math
coursework, or trolls.
Once upon a time, I would have believed that :D
What happened? Do you now expect to divide by zero and obtain a unicorn?
Maybe I don't understand your reply.
Doug
2017-07-26 00:40:25 UTC
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Post by Bill
Post by Doug
Post by Doug
Post by Doug
Thanks, yes I did notice that, it is a slightly
different symbol. I think the Riemann Sphere is a
partial attempt at solving the problem, but really
leaves the same fundamental question hanging.
Think of the graph of y=1/x, or y=1/z. There is
no
Post by Doug
Post by Doug
fundamental question
hanging--at least not for anyone who has done any
sort of math
coursework, or trolls.
Once upon a time, I would have believed that :D
What happened? Do you now expect to divide by zero
and obtain a unicorn?
Maybe I don't understand your reply.
lol yes you do understand it, pretty much.

It is my suspicion that there is an actual answer to n/0, it just has escaped us so far. In other words, once upon a time I believed in the fairy tale that we live at the pinnacle of knowledge and advancement. Now, I know better :)
Julio Di Egidio
2017-07-26 01:33:38 UTC
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Post by Doug
It is my suspicion that there is an actual answer to n/0, it just has escaped us so far. In other words, once upon a time I believed in the fairy tale that we live at the pinnacle of knowledge and advancement. Now, I know better :)
You just keep pushing around this bullshit that "we don't know" despite few
here by now do have explained what there is to explain.

So welcome to my kill-file, under the rubric yet another fucking spammer.

*Plonk*

Julio
conway
2017-07-26 02:49:43 UTC
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All numbers are composed of value and space. value is a given quantity of existence other than space. space is a given quantity of dimension. the value of zero is zero. the space of zero is 1. ......

multiplication and division by zero solved for.....
Doug
2017-07-26 15:49:37 UTC
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Post by conway
All numbers are composed of value and space. value
is a given quantity of existence other than space.
space is a given quantity of dimension. the value
e of zero is zero. the space of zero is 1. ......
multiplication and division by zero solved for.....
Why can't dimension be zero? zero width, zero height, etc...
Doug
2017-07-26 16:02:44 UTC
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On Wednesday, July 26, 2017 at 3:20:13 AM UTC+2, Doug
Post by Doug
It is my suspicion that there is an actual answer
to n/0, it just has escaped us so far. In other
words, once upon a time I believed in the fairy tale
that we live at the pinnacle of knowledge and
advancement. Now, I know better :)
You just keep pushing around this bullshit that "we
don't know" despite few
here by now do have explained what there is to
explain.
So welcome to my kill-file, under the rubric yet
another fucking spammer.
*Plonk*
Julio
I appreciate very much all the answers I have received, even yours.

I'm certainly not spamming.

I sincerely believe that there is something more to be discovered on this subject, and I think it is important to discuss such things. I am also interested to hear what people have to say, and to learn what has already been published by people with far more education than myself.

Considering the fact that this subject opens up a huge amount of philosophical discussion and debate, I find it surprising that anyone would suggest it has been "explained".

Are you annoyed that I don't offer my own thoughts on the subject, other than the "we don't know" thing? I doubt anyone cares about my own thoughts, and I didn't start this discussion to debate people. I am confident that I have an answer. I just wanted to know if it is already commonly known. If that were the case, it certainly would have come up by now in the present discussion, so I can be confident that it is for some strange reason still unknown generally.
Peter Percival
2017-07-26 16:29:32 UTC
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[...] I am confident that I have an answer. I just wanted to know if
it is already commonly known. If that were the case, it certainly
would have come up by now in the present discussion, so I can be
confident that it is for some strange reason still unknown
generally.
What is your answer?
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
conway
2017-07-26 16:29:15 UTC
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Doug

Have you ever seen zero height, zero width? It does not exist. Even on the quantum level. Any dimension....is not nothing.....nothing does not exist. Zero is nothing more than a dimension with the absence of value.
Doug
2017-07-26 23:14:32 UTC
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Post by Doug
[...] I am confident that I have an answer. I just
wanted to know if
it is already commonly known. If that were the
case, it certainly
would have come up by now in the present
discussion, so I can be
confident that it is for some strange reason still
unknown
generally.
What is your answer?
--
Do, as a concession to my poor wits, Lord Darlington,
just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be
intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's
Fan
I'm going to write something up and post it someplace. I'll link it when I'm done.

No doubt I will suffer a lot of abuse afterwards :D
conway
2017-07-27 00:25:36 UTC
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Doug...

I am very interested in this conversation with you. I can provide extensive knowledge on this matter. Both fact, and theoretical. Please...should you wish to continue....I would talk with you at length (on this matter)....

Also....most people on this forum are literal trolls. They must be ignored or risk emotional defeat in the matter.
Doug
2017-07-26 23:16:13 UTC
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Post by conway
Doug
Have you ever seen zero height, zero width? It does
not exist. Even on the quantum level. Any
dimension....is not nothing.....nothing does not
exist. Zero is nothing more than a dimension with
the absence of value.
Gotcha. So, probably the same is true of infinity? The two are just imaginary constructs?
Jonathan Cender
2017-07-27 04:01:50 UTC
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Are you familiar with the idea in Euclid's Elements that "point" is without width, height, or depth? How does this compare with your wondering

"> Why can't dimension be zero? zero width, zero height, etc..."
conway
2017-07-27 13:30:43 UTC
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Lol I have never heard of such.....I have never seen a point without dimension.....I can not represent a point with out the use of dimension.....lol....to say a point has no dimension is equivalent to saying the sun has no warmth......please provide link to show Euclid claim......lol
conway
2017-07-27 13:36:41 UTC
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John.

So Euclid did make such claim, latter to retract it.

http://aleph0.clarku.edu/~djoyce/elements/bookI/defI1.html

The description of a point, “that which has no part,” indicates that Euclid will be treating a point as having no width, length, or breadth, but as an indivisible location.

Can something be in a "location" yet not have no dimension.....it can not... such terms are contradictory. His point here was to say that "point" can not be further reduced.....not that it lacked dimension....again this he latter clarifies.
Doug
2017-07-28 11:44:58 UTC
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Post by Jonathan Cender
Are you familiar with the idea in Euclid's Elements
that "point" is without width, height, or depth? How
does this compare with your wondering
"> Why can't dimension be zero? zero width, zero
height, etc..."
Yes. My understanding is that electrons, for example, appear to have no physical dimension.
Doug
2017-07-28 11:46:27 UTC
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Post by conway
John.
So Euclid did make such claim, latter to retract it.
http://aleph0.clarku.edu/~djoyce/elements/bookI/defI1.
html
The description of a point, “that which has no
o part,” indicates that Euclid will be treating a
point as having no width, length, or breadth, but as
an indivisible location.
Can something be in a "location" yet not have no
dimension.....it can not... such terms are
contradictory. His point here was to say that
"point" can not be further reduced.....not that it
lacked dimension....again this he latter clarifies.
I can't help but say "I see your point" ;)
Jonathan Cender
2017-07-29 02:45:12 UTC
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I would be very interested to know where Euclid retracted or amended this claim about "point."
I will look through later, but I would also appreciate assistance:)

BTW. If you don't know, David Joyce, the editor of the version of Euclid's Elements you linked to, is quite active on Quora - for example - https://www.quora.com/What-philosophers-criticized-math-Why
Post by conway
So Euclid did make such claim, latter to retract it.
conway
2017-07-29 03:41:36 UTC
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Jonathan

Sure!.....

https://en.wikipedia.org/wiki/Point_(geometry)

Quote from link

"However, Euclid's postulation of points was neither complete nor definitive, and he occasionally assumed facts about points that did not follow directly from his axioms, such as the ordering of points on the line or the existence of specific points. In spite of this, modern expansions of the system serve to remove these assumptions."

I wrongly implied that Euclid retracted it. It was not that it was retracted "per say" by Euclid....it was that it was adapted and updated with the work of mathematicians to follow.

Consider.....

If a point has no real dimension, than how can a collection of points (all of which) have no dimension, create a line, which clearly has dimension

Consider......

How can a "point" exist is a Euclidian space, (space) being dimension, yet it's self hold no dimension

This is like saying there is salt in my water, but the salt therein is not in the water.

Quote from link......

"A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space."

I can not have a location, if I first do not have a dimension. All dimension, rest in a location.....

Also

0 is not nothing. So to say 0 dimension, does NOT mean, absence of dimension.....only undefined, or unreducible, or absent parts.....
FredJeffries
2017-07-29 15:11:57 UTC
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Post by conway
Consider.....
If a point has no real dimension, than how can a collection of points (all of which) have no dimension, create a line, which clearly has dimension
Teamwork

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