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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

#211

Post by dwohnitmok »

CamperBob2 wrote: post #202
dwohnitmok wrote: post #154

Earlier quoted context omitted.

Forcing is about a completely different question. The question is not where there are reals between reals, it's a question of the size of sets. In particular is there a set strictly larger than the natural numbers, but strictly smaller than the reals? Forcing allows us to construct such a set in ZFC.

Not a mathematician, so this probably has an obvious answer, but who says that "size" is a necessary universal property of a set? It doesn't seem any more rational to speak of size as a required property of a set than it would be to treat color or weight that way. Some sets have those attributes, but not all. It seems perfectly reasonable to say that the set of real numbers is one of those sets that doesn't have a "s…

Size is a hand-wavy way of saying it. What we're really asking is given two sets A and B, is there a function from A to B such that no two elements from A are mapped to a single element in B?

It turns out if you use ZFC (in particular choice) then one direction must hold. Either such a function exists from A to B or from B to A. This means that we can order all sets using the existence of these functions, and so in that sense size is a "universal property."

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

#212

Post by dwohnitmok »

red_trumpet wrote: post #177
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…

Thanks for that perspective! Searching around I found this article https://plato.stanford.edu/entries/platonism-mathematics/#Tr... which even distinguishes mathematical Platonism and truth-value realism. Interesting stuff!

Indeed, that article is correct. More accurately I should be calling this position mathematical realism (although the lines between the two can get a bit blurry).

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

#213

Post by rssoconnor »

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…

It's not "okay" because such a system proves that various particular Turing machines halt when, in fact, those machines do not halt. See https://news.ycombinator.com/item?id=27847719.

But to be a bit more specific, ¬Con(ZFC) says that the Turing machine that searches for a contradiction in ZFC does indeed halt. However (in all likelihood) such a machine does not actually halt, in the sense that it does not halt in 1 step, and it does not halt in 2 steps, and it does not halt in 3 steps, etc., and indeed (in all likelihood) for each numeral n, ZFC even proves that the machine does not halt in n steps.

(Now there is a small possibility that maybe such a machine does halt in some particular number of steps. If it does actually halt, that means it has found a proof that ZFC is inconsistent. But this scenario is even worse, because it means that ZFC itself is not only unsound, but inconsistent, (and hence ZFC+¬Con(ZFC) is also inconsistent). In particular ZFC+¬Con(ZFC) being inconsistent means it proves that every Turing machine halts, which is even more wrong in general, even if it happens to be right about this particular machine.)

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

#214

Post by dwohnitmok »

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.

It is a significant position among mathematicans and logicians who study the foundations of mathematics.

It is probably not how most non-foundations mathematicians think of mathematics (especially larger infinities).

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

#215

Post by drdeca »

dexwiz wrote: post #136

I'm no mathematician, but I have always found it strange infinities are talked about as physical states (towers of tall towers, etc), and not functions. Integer number counting is essentially a successor function; take N, add 1, output N+1. One input, one output. Real number counting is a bit more loose. To find the numbers between .1 and .2, you find all fractionals of a given size, normally 1/10. So we get .10, .11…

There is a way of mapping different ordinal numbers which are less than some large countable ordinal (idr how high up it goes. I suspect it isn’t well defined at least past the church kleene ordinal, but I think it is much before that, but also I could be totally wrong) to different functions from integers to integers which increase at different rates.

These methods include the Hardy hierarchy, and the fast growing hierarchy.

(But regarding all infinities being equal, that’s just incorrect, sorry.)

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

#216

Post by skissane »

rssoconnor wrote: post #92
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…

Dropping down to Peano Arithmetic for a moment. We can consider adding a new constant 'c' for a natural number to the language along with the following infinite list of axioms about this remarkable constant: - 0 - 1 - 2 - 3 ... Adding all these axioms is consistent. I.e. you can do induction upto 'c', whatever it is. Why is it consistent? Because if there was a contradiction, the proof of such a contradiction would b…

> in that it yields only models that have elements that are larger than every written numeral.

Are these like hyperintegers (or should I say hypernaturals), as in the hyperrreal numbers used in non-standard analysis?

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

#217

Post by myWindoonn »

sharpener wrote: post #184

The mileage of others may vary, but it has been my experience that there is no cogent solid proof of uncountability that can withstand concerted critique.[0] Being charitable one might argue that the meanings of terminology had been lost in translation over time and that perhaps Cantor was trying to create non-standard analysis, but then the diagonal argument seems to represent nothing more than the truism that finit…

Hi, Lawvere pummelled your position into the ground a while ago: http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf Your critique involves repeatedly crossing the boundary between the inside and outside of the system in question; Lawvere works entirely inside the system, and shows that the paradoxes of self-reference arise from our interpretations. https://arxiv.org/abs/math/0305282v1 explains with many examples.

Hi downvoters: Use your words when somebody is wrong. Your downvotes aren't helpful here for finding the truth.

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

#218

Post by myWindoonn »

chmod775 wrote: post #135
btilly wrote: post #88

As a constructivist I'll be over in the corner that says that there are only a countable number of real numbers, and the unimaginable number of unimaginable infinities that classical mathematics insiste exists is all made up nonsense. That, in fact, things that can't ever be named, even in principle, don't actually exist. What is interesting is that as shocking as constructivism may be, there is no logical flaw in it…

> there are only a countable number of real numbers Then you should be able to come up with a function that assigns a natural number uniquely to each real number. Of course if you tried that I could immediately name you a real number, or a pair of them, for which your rule doesn't work.

Turing machines are countable. In some constructive settings, every real number is computable, as an axiom; those settings allow us to talk about the Turing machines which compute a particular real number.

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

#219

Post by mmmBacon »

TameAntelope wrote: post #207
twelfthnight wrote: post #153

Earlier quoted context omitted.

Here's some background on the proof [1]. Here's a video explaining it little better [2]. [1] https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument [2] https://www.youtube.com/watch?v=elvOZm0d4H0

I guess I was focusing on the word "easy", heh.

You do this in sophomore level Real Analysis. So as far as pure math goes, easy.

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

#220

Post by a1369209993 »

rssoconnor wrote: post #213
Someone wrote: post #199

Earlier quoted context omitted.

> 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…

It's not "okay" because such a system proves that various particular Turing machines halt when, in fact, those machines do not halt. See https://news.ycombinator.com/item?id=27847719 . But to be a bit more specific, ¬Con(ZFC) says that the Turing machine that searches for a contradiction in ZFC does indeed halt. However (in all likelihood) such a machine does not actually halt, in the sense that it does not halt in 1…

> such a system proves that various particular Turing machines halt when, []in fact[], those machines do not halt.

Er, no. The fact is that there is no fact of the matter as to whether those particular Turing machines either a: do not halt at all, or b: halt after a (colloquially) infinite number of steps. (A implication of there being no fact of the matter is that, empirically, we can't tell the difference by running them, but we can't tell the difference for a machine that halts in 2^(2^(2^256)) steps (or not at all) either, so that's not very interesting on it's own.)

(As you note, we don't actually know that (for example) a Turing machines looking for contradictions in ZFC is one of those particular machines; indeed I'm not sure offhand if we actually know of any specific example of such a machine. But that's presumably not the issue here.)

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