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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

#251

Post by oehtXRwMkIs »

saithound wrote: post #247
zenon wrote: post #237

Earlier quoted context omitted.

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.

You're looking at the result for aesthetic value not Platonism.

And by the measure of "most" meaning majority it would still be most.

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

#252

Post by TuringTest »

HelloNurse wrote: post #122
TuringTest wrote: post #100

Earlier quoted context omitted.

> The same applies to rational numbers (which exist more than real numbers) Wait, did you say there are more rationals than reals? Isn't that the other way around? I don't know if that's a slip of the tongue or I'm missing something in my recall of basic math lessons

Real numbers do not exist: they are convenient, but there is a high risk of crossing over from reality-relevant math to nonsense because of infinite calculations. Real numbers are of course more numerous than rational numbers, just like unicorn horns are more numerous than horse horns, but it's an entirely different question.

No numbers exist, they are abstraction on human-created procedures. Even naturals, which are an abstraction over pointing to things and counting.

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

#253

Post by TuringTest »

gowld wrote: post #179
TuringTest wrote: post #100

Earlier quoted context omitted.

> The same applies to rational numbers (which exist more than real numbers) Wait, did you say there are more rationals than reals? Isn't that the other way around? I don't know if that's a slip of the tongue or I'm missing something in my recall of basic math lessons

They are "more existent" ("less fictional"), not "more in cardinality".

How are they more existent? They are both patterns of recognized concepts in human brains.

Is it because more people understand rationals than reals? Ok, that would make rationals more existent, as they exist in more brains.

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

#254

Post by pwdisswordfish8 »

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.

What does ⅓ map to?

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

#255

Post by hutzlibu »

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 claim that the set of natural numbers has the same size as that of the rational numbers"

Where can I read more about that? Because, both are infinite, but there still should be more rational numbers, than natural numbers?

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

#256

Post by jari_mustonen »

hutzlibu wrote: post #255
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…

"the claim that the set of natural numbers has the same size as that of the rational numbers" Where can I read more about that? Because, both are infinite, but there still should be more rational numbers, than natural numbers?

You can count rational numbers and everything countable is the size of infinity as natural numbers.

Google ”counting rational numbers” and ”different sizes of infinity” to learn more.

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

#257

Post by skoodge »

pdonis wrote: post #205
skoodge wrote: post #157

Earlier quoted context omitted.

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.

Sure, that is why I wrote:

> In such a sense then, we can trivially "count" the real numbers unless we hold the philosophical view that there are real numbers that are not expressible. This is where it becomes a question of philosophy of mathematics, not mathematics proper.

I'm not disagreeing with your interpretation of Cantor's diagonal proof, I'm merely pointing out that this interpretation depends on a very specific philosophical view of mathematics, namely the platonist view that the real numbers exist independently from their expressions in any natural or formal language and that it makes sense to say that there are real numbers that are not expressible.

And yeah, nearly all working mathematicians will agree with this view and from their perspective the real numbers are uncountable, period, and you are right that what I sketched "doesn't work".

But I think it's important to remember that there are or could be alternative philosophical views of mathematics that lead to a different interpretation, which will reject not the mathematical validity of Cantor's diagonal proof, but rather its usefulness or relevance. After all, how can you convince someone that there are real numbers that are not expressible? By their very nature they cannot be practically used in any calculation, so how could you convince someone who is not convinced by this philosophical assumption of Cantor's diagonal proof?

In other words, Cantor's diagonal proof cannot prove that there are real numbers that are not expressible, because the proof only makes (philosophical) sense if you accept this viewpoint in the first place.

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

#258

Post by quisnam »

hutzlibu wrote: post #255
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…

"the claim that the set of natural numbers has the same size as that of the rational numbers" Where can I read more about that? Because, both are infinite, but there still should be more rational numbers, than natural numbers?

Many entry-level real analysis courses will cover cardinality after introducing sets and functions. There's also a short article on Wikipedia. [0]

Your intuition about there being more rational numbers might be based on viewing the rationals as a proper superset of the natural numbers. You might similarly consider that there are more natural numbers than even natural numbers. However, by "renaming" every even number, and that shouldn't change how many there are, to half its value, we obtain the natural numbers. Formally, there exists a bijection between N and 2N, as between N and Q, and this is what mathematicians mean when they say that sets have the same size, or cardinality.

[0]: https://en.wikipedia.org/wiki/Cardinality

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

#259

Post by chmod775 »

skoodge wrote: post #257
pdonis wrote: post #205

Earlier quoted context omitted.

> 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.

Sure, that is why I wrote: > In such a sense then, we can trivially "count" the real numbers unless we hold the philosophical view that there are real numbers that are not expressible. This is where it becomes a question of philosophy of mathematics, not mathematics proper. I'm not disagreeing with your interpretation of Cantor's diagonal proof, I'm merely pointing out that this interpretation depends on a very speci…

> 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.

You could say the same about just real numbers, which can also be ordered from smallest to largest. This definitely not 'trivially' implies countability.

Edit: I edited my post shortly after posting from "Humor me and count out the first two real numbers".

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

#260

Post by sharpener »

curtisf wrote: post #224
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…

This presents a confused understanding of Cantor's diagonalization argument. You are shrouding in complexity something that is straightforward. The complete proof of distinct infinite cardinalities can be stated succinctly and clearly in only a few lines, without referencing the reals at all. You don't need to vaguely refer to "four steps", you should precisely elaborate the steps of the proof you view as problematic…

> Define a function z(n) = 1 - f(n)(n).

I don't understand the notation f(n)(n). Is it related to f_{nn} in LaTeX notation? Your later text suggests maybe it was aiming at f(n,n) so I will assume that.

I recognise a form of this argument and I might have tackled it in the supplementary materials I created that are referenced in the article. Let me know.

> However, z(k) = 1 - f(k)(k). Yet f(k) = z, so z(k) = 1 - z(k).

I'm assuming this was intended to be: z(k) = 1 - f(k). Yet f(k) = z, so z(k) = 1 - z(k).

For some k, z(k) = 0.5. f(k) = 0.5. Seems Ok.

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