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How many real numbers exist? New proof moves closer to an answer

quantamagazine.org

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Re: How many real numbers exist? New proof moves closer to an answer

#241

Post by logicchains »

kmill wrote: post #91
tgflynn wrote: post #72

Earlier quoted context omitted.

> he resulting system is unsound (in the sense of Tarski) because it asserts the existence of natural numbers that have no "written form" (i.e. the existance of natural numbers that are larger than any term you can write to denote a natural number). Isn't that basically the definition of the natural numbers, ie. if you write down any natural number (say n) I can always construct a natural number that is larger than i…

This surprisingly doesn't mean repeatedly adding 1 will exhaust all natural numbers -- there are models for the natural numbers with elements that can't be reached this way! The ultrafilter construction gives one such model. You take the set of all sequences of natural numbers (n1, n2, n3, ...) then use an ultrafilter to decide which of these sequences are considered to be equal. The usual operations of natural numbe…

>The ultrafilter construction gives one such model.

It's worth noting that this construction relies on the axiom of choice, so exists in ZFC but not ZF. Generally a lot of counter-intuitive constructions disappear when eliminating the axiom of choice (such as the Banach–Tarski paradox and non-continuous functions).

Re: How many real numbers exist? New proof moves closer to an answer

#242

Post by kmill »

kmill wrote: post #234
R0b0t1 wrote: post #233

Earlier quoted context omitted.

Can you explain what use these have? If natural numbers are (n, n, n, ...) then you have just made a new type of number not comparable to natural numbers.

A theoretical use is that it shows that the axioms of the natural numbers (the Peano axioms) aren't enough to pin down what we think the natural numbers should be -- there are these crazy non-standard natural numbers out there! In context of the discussion, this non-standard model of the natural numbers gives some intuition about what happens if the consistency of ZFC is independent of ZFC and you add in the axiom th…

> calculate the points (like in the end of my comment)

Oh, I forgot I commented more than once today on HN. I was referring to this one: https://news.ycombinator.com/item?id=27851188

Re: How many real numbers exist? New proof moves closer to an answer

#243

Post by [deleted] »

plus wrote: post #188
phkahler wrote: post #163

You can map all the real numbers to the interval 0 0.0 => 0 0.1 => 1 0.2 => 2 ... 0.14159 => 95141 The argument against this is that there are real numbers with an infinite number of digits, which will not have a specific integer associated with them. Or do they? Can there be an "infinitely large integer?" or are there just infinitely many of them? I think this question gets to the core of the problem.

> The argument against this is that there are real numbers with an infinite number of digits, which will not have a specific integer associated with them. Or do they? If it was possible to construct your mapping, then there would be a well-defined sorting of the reals between 0 and 1 based on their integer representation (e.g. we could sort the set {0.05, 0.1, 0.2} => {50, 1, 2} to [0.1, 0.2, 0.05] => [1, 2, 50]). Ho…

[deleted]

Re: How many real numbers exist? New proof moves closer to an answer

#244

Post by DangitBobby »

plus wrote: post #188
phkahler wrote: post #163

You can map all the real numbers to the interval 0 0.0 => 0 0.1 => 1 0.2 => 2 ... 0.14159 => 95141 The argument against this is that there are real numbers with an infinite number of digits, which will not have a specific integer associated with them. Or do they? Can there be an "infinitely large integer?" or are there just infinitely many of them? I think this question gets to the core of the problem.

> The argument against this is that there are real numbers with an infinite number of digits, which will not have a specific integer associated with them. Or do they? If it was possible to construct your mapping, then there would be a well-defined sorting of the reals between 0 and 1 based on their integer representation (e.g. we could sort the set {0.05, 0.1, 0.2} => {50, 1, 2} to [0.1, 0.2, 0.05] => [1, 2, 50]). Ho…

> then there would be a well-defined sorting of the reals between 0 and 1 based on their integer representation

Why is that?

Re: How many real numbers exist? New proof moves closer to an answer

#245

Post by kmill »

logicchains wrote: post #241
kmill wrote: post #91

Earlier quoted context omitted.

This surprisingly doesn't mean repeatedly adding 1 will exhaust all natural numbers -- there are models for the natural numbers with elements that can't be reached this way! The ultrafilter construction gives one such model. You take the set of all sequences of natural numbers (n1, n2, n3, ...) then use an ultrafilter to decide which of these sequences are considered to be equal. The usual operations of natural numbe…

>The ultrafilter construction gives one such model. It's worth noting that this construction relies on the axiom of choice, so exists in ZFC but not ZF. Generally a lot of counter-intuitive constructions disappear when eliminating the axiom of choice (such as the Banach–Tarski paradox and non-continuous functions).

> non-continuous functions

I think the step function doesn't depend on choice. Say f(x) = 0 if x = 0. Also, suppose we define the reals through Dedekind cuts. We can define f(x) by checking whether 0 is an element of x as a cut. We can prove f is discontinuous in the usual way.

What I had heard is that Brouwer proved, using some kind of constructive/intuitionistic logic, that every function is continuous, but ZF is still based in classical logic. I believe the step function is not definable constructively.

Re: How many real numbers exist? New proof moves closer to an answer

#247

Post by saithound »

zenon wrote: post #237
zajio1am wrote: post #206

Earlier quoted context omitted.

> this entire article is implicitly assuming a Platonist philosophical foundation Is mathematical platonism still significant position between mathematicians? I thought it is outdated since Lobachevsky.

In this survey, most "Philosophers of mathematics" endorsed Platonism: https://philpapers.org/surveys/results.pl?affil=Target+facul...

edit: I did not pay enough attention, so what I wrote is wrong, see the comment below

In the survey you linked, it looks like 45.7% of the surveyed philosophers of mathematics endorse Platonism. That's a lot, but not "most", by any measure.

Re: How many real numbers exist? New proof moves closer to an answer

#248

Post by graycat »

Someone wrote: post #199
rssoconnor wrote: post #65

Earlier quoted context omitted.

> but at the same time there is no "right answer"—all of these axioms, after all, are independent of ZFC, and it's "ok" to add any of them. Minor quible: Just because a potential axiom is independent of ZF(C) doesn't make it necessarily "okay" to add. Potential axioms can be unsound, for example if they prove new / untrue Sigma_1 statements. As an example, even in the likely circumstance that ¬Con(ZFC) is independent…

> because it asserts the existence of natural numbers that have no "written form" I don’t see why that should imply it wouldn't be "okay" to add ¬Con(ZFC). It may be highly counterintuitive, but the history of mathematics is full of counterintuitive results that nowadays are accepted as true in mainstream mathematics. Well-known examples are the existence of irrational numbers, the claim that the set of natural numbe…

The Banach-Tarski paradox is fine with me: Not all subsets of the real line or R^n for the set of real numbers R and positive integer n are measurable. Yes, the usual example of a non-measurable set uses the axiom of choice. The sets in the Banach-Tarski paradox are not measurable -- okay.

Of course the natural numbers can be put into 1-1 correspondence with the rationals -- how to show that is the classic Cantor diagonal argument.

At a tea before a seminar, I asked Max Zorn what Paul Cohen had just proved. Zorn didn't explain it and instead just loaned me his copy of Cohen's paper -- I lost it in a move! If Zorn wasn't strongly interested in Cohen's proof, then neither should I be.

For any set of axioms, there will be statements we can neither prove nor disprove. Surprising, interesting, got it.

Zermelo-Fraenkel set theory with the axiom of choice seem fine to me -- now back to the real work!

Re: How many real numbers exist? New proof moves closer to an answer

#250

Post by saithound »

zajio1am wrote: post #206
dwohnitmok wrote: post #121

Man there's a lot of juicy stuff in this article (Woodin's Ultimate L program gets briefly alluded to at the end of the article). I just want to point out, because the HN crowd seems to generally not be mathematical Platonists, that this entire article is implicitly assuming a Platonist philosophical foundation. This may cause confusion for lay readers who are not mathematical Platonists. In other words the article a…

> this entire article is implicitly assuming a Platonist philosophical foundation Is mathematical platonism still significant position between mathematicians? I thought it is outdated since Lobachevsky.

Note that a view can be a majority position even if it is extremely outdated. This often happens when the outdated position is much easier to explain than the more nuanced alternatives. E.g. when surveyed, evangelical preachers often endorse Arianism, despite it being considered outdated since 325AD.
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