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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

#201

Post by rkagerer »

ncmncm wrote: post #192

> How many real numbers exist? You might be tempted to say "lots". And you would be right, as far as that goes. But that doesn't satisfy a real mathematician. The question that immediately arises is whether "lots" is "enough". And that leads the better sort of mathematician inevitably to: "enough for what?" That is what mathematicians are deep in the middle of exploring, now. For example, when you are asked, "Does th…

"Does this skirt make my butt look too big?"

"Too big for what?"

Yeah let me know how that works out.

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

#202

Post by CamperBob2 »

dwohnitmok wrote: post #154
finnh wrote: post #120

Earlier quoted context omitted.

Great question, I have no answer. The article explained forcing in such a way as to simply restate what I thought we already knew: given a real, there is no "next" real. (ie, there are a non-countable-infinite number of reals between any two reals). I don't see the newness that forcing brings to this.

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 "size" property.

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

#203

Post by pdonis »

btilly wrote: post #118
scapp wrote: post #105

Earlier quoted context omitted.

You sound like you need to read this [0] answer to the question "Are real numbers countable in constructive mathematics?". > You are using the word "constructive" in an unusual way. It is true that, in ZFC, the set of computable real numbers is countable, but that is not directly a statement about constructive mathematics. > Not every school of constructive mathematics identifies real numbers with algorithms; that's…

Yes, there are multiple constructivist approaches possible. However since my objection to classical approaches is that I want "X exists" to be meaningful, I like mathematical objects that can be written down with a finite number of symbols in a finite space. Which means that I'm only interested in a countable universe of possible mathematical things. If you say "exists" about anything else, I'll understand you - I do…

> I like mathematical objects that can be written down with a finite number of symbols in a finite space.

Which is fine, but I don't think it justifies the claim that there are only countably many real numbers. The only claim it justifies is that there are only a finite (not even countable, since "countable" implies infinitely many) set of numbers that you find useful.

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

#204

Post by pdonis »

btilly wrote: post #193
LudwigNagasena wrote: post #139

Earlier quoted context omitted.

What’s the problem with it being “artificial”? Is your problem purely linguistic? You just dislike the word “exists” being used in this context?

The problem is that exists comes to mean something technical that doesn't match common usage. Let's take my favorite example. In graph theory, a minor of a graph is a graph you can get by removing vertices, removing edges, or by replacing an edge-vertex-edge triple with a single edge. Many categories of graphs are closed under the act of taking minors. For example planar graphs, graphs you can draw on the plane with…

These concerns don't apply to the claim that the set of real numbers is uncountable. Cantor's diagonal proof is constructive: given any countable set of real numbers, it tells you how to construct a real number that is not in the set. That is sufficient to show that the set of real numbers cannot be countable. Also, even though many real numbers cannot be written down with a finite set of symbols, Cantor's diagonal proof can be.

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

#205

Post by pdonis »

skoodge wrote: post #157
chmod775 wrote: post #135

Earlier quoted context omitted.

> 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.

That depends on what you mean by "assigns uniquely", "rule" and "doesn't work", which is why this question is deeply entangled with philosophical issues that cannot be settled purely mathematically. It is obvious that all expressions in the English language can be ordered from smallest to largest and lexicographically, which makes these expressions trivially countable. We can thus assign natural numbers to real numbe…

> We can thus assign natural numbers to real numbers by assigning numbers to their expressions in a natural or formal language

This doesn't work because not all real numbers have expressions in a natural or formal language. This is easily shown by an obvious variation on Cantor's diagonal proof, applied to your lexicographically ordered list of expressions in any natural or formal language.

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

#206

Post by zajio1am »

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.

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

#207

Post by TameAntelope »

twelfthnight wrote: post #153
TameAntelope wrote: post #140

Earlier quoted context omitted.

How can you easily prove that the set of real numbers is not countable? I don't think it's as easy as you claim, but I'm kind of a dummy so it's quite probably I'm wrong.

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.

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

#208

Post by LudwigNagasena »

btilly wrote: post #193
LudwigNagasena wrote: post #139

Earlier quoted context omitted.

What’s the problem with it being “artificial”? Is your problem purely linguistic? You just dislike the word “exists” being used in this context?

The problem is that exists comes to mean something technical that doesn't match common usage. Let's take my favorite example. In graph theory, a minor of a graph is a graph you can get by removing vertices, removing edges, or by replacing an edge-vertex-edge triple with a single edge. Many categories of graphs are closed under the act of taking minors. For example planar graphs, graphs you can draw on the plane with…

In the usual sense. I don’t see a problem. Just because you don’t have a perfect knowledge of something, it doesn’t mean that the thing isn’t real.

I am not sure where this idea even comes from? To be honest, this sounds completely ridiculous.

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

#209

Post by whoaisme »

ncmncm wrote: post #192

> How many real numbers exist? You might be tempted to say "lots". And you would be right, as far as that goes. But that doesn't satisfy a real mathematician. The question that immediately arises is whether "lots" is "enough". And that leads the better sort of mathematician inevitably to: "enough for what?" That is what mathematicians are deep in the middle of exploring, now. For example, when you are asked, "Does th…

Ken M is that you?

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

#210

Post by drdeca »

ssivark wrote: post #55
jaywalk wrote: post #8

I'm not trying to be flippant, although it may come off that way: why does any of this matter?

Let me venture a possible application, since there are already many responses celebrating pure mathematics. As far as we understand, the natural numbers are not sufficient for modeling physical phenomena. The reals/complex while immensely useful also occasionally turn out to be “too complicated” to give theoretical guarantees/proofs of models working well. On the practical side, that means that these models/algorithm…

Whether the cardinality of the reals is aleph 1, aleph 2, or something larger, is independent of ZFC.

In a model of ZFC in which the set of reals has cardinality greater than aleph_1 , I wouldn’t be surprised if there is a subfield of the reals of cardinality aleph_1 , but I would be surprised if such a field was useful for things like that. Such a field would, of course, not be complete with respect to the usual metric on the rationals, so we wouldn’t have the desired convergence properties. We wouldn’t really be able to do infinite sums in it? (Well, perhaps some other sense of infinite summation could be done, but it wouldn’t be the usual sense.)

In addition, because such a subfield would only exist as an uncountable proper subfield in some models of ZFC, I find it hard to imagine that it would allow computations that wouldn’t otherwise work? I suppose it could motivate some computations which would then also work regardless of what model of ZFC is being used / is true ?

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