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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#21

Post by upquark »

FabHK wrote: post #14
upquark wrote: post #12

Earlier quoted context omitted.

"Real numbers that cannot be defined with language" is not a well-defined set though. Just like "integers that cannot be described in less than 100 words" is not well-defined (I could reach a contradiction by pointing to the "smallest integer that cannot be described in less than 100 words").

True, but the argument that one set is larger still works.

Wait, you're agreeing with me that the set in question is ill-defined, but still somehow comparable to other sets? I am not sure I follow. My statement is that you cannot meaningfully talk about "all the numbers that cannot be described with language". If you could, I'd ask you if this set intersected with [0, 1] has a lower bound / 'inf' that's contained in it, and if so, did I just describe that lower bound with language? I'm sure some sort of paradox similar to the one with integers can be constructed here...

In my view, every real number is well-defined and there's nothing controversial about the set of real numbers. If the infinite aspect of it causes some researchers to call it a "mathematical fantasy", so be it, so is literally every other mathematical model we use in our lives.

Re: How real are real numbers? (2004)

#22

Post by btilly »

surement wrote: post #5
whatshisface wrote: post #3

"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.) I think Borel has confused names with things. The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never…

> The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never write expressions for Yes it does: there are countably many numbers we can define, and the reals are uncountable, therefore there are uncountably many numbers that cannot be defined.

Whether or not it does is a matter of which philosophy you choose.

A Platonist may believe that those numbers exist in an abstract space we cannot reach. A Formalist simply defines "existence" in such a way that this is true without worrying about whether they really exist. And a Constructivist denies the existence of things that cannot actually be written down, at least in theory.

Re: How real are real numbers? (2004)

#23

Post by robinhouston »

whatshisface wrote: post #3

"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.) I think Borel has confused names with things. The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never…

A rather more technical – but very interesting – take on this question is [0], which opens:

> One occasionally hears the argument — let us call it the math-tea argument, for perhaps it is heard at a good math tea — that there must be real numbers that we cannot describe or define, because there are are only countably many definitions, but uncountably many reals. Does it withstand scrutiny?

then explains in detail why it does not withstand scrutiny.

[0] https://arxiv.org/abs/1105.4597

Re: How real are real numbers? (2004)

#24

Post by Smaug123 »

upquark wrote: post #21
FabHK wrote: post #14

Earlier quoted context omitted.

True, but the argument that one set is larger still works.

Wait, you're agreeing with me that the set in question is ill-defined, but still somehow comparable to other sets? I am not sure I follow. My statement is that you cannot meaningfully talk about "all the numbers that cannot be described with language". If you could, I'd ask you if this set intersected with [0, 1] has a lower bound / 'inf' that's contained in it, and if so, did I just describe that lower bound with la…

Your example is faulty.

> I'd ask you if this set intersected with [0, 1] has a lower bound / 'inf' that's contained in it, and if so, did I just describe that lower bound with language?

The "indescribable numbers" are dense in [0,1], and so (if the set exists) the inf of the set of indescribable numbers which are between 0 and 1 is 0. Perfectly describable.

Re: How real are real numbers? (2004)

#25

Post by FabHK »

upquark wrote: post #21
FabHK wrote: post #14

Earlier quoted context omitted.

True, but the argument that one set is larger still works.

Wait, you're agreeing with me that the set in question is ill-defined, but still somehow comparable to other sets? I am not sure I follow. My statement is that you cannot meaningfully talk about "all the numbers that cannot be described with language". If you could, I'd ask you if this set intersected with [0, 1] has a lower bound / 'inf' that's contained in it, and if so, did I just describe that lower bound with la…

There are more people on earth than, say, kings. That's true, even though I can't enumerate all non-kings, and even if the set of non-kings is somehow ill-defined.

Re: How real are real numbers? (2004)

#26

Post by dbcurtis »

1. Given any two real numbers on the real number line, you can find another real number between those two points.

2. The Planck length is the smallest unit of distance with any meaning.

3. The universe has finite diameter.

Discuss.

4. For extra credit: Given 2 and 3, above, it follows that both the diameter and circumference of the universe can be expressed in Planck lengths as integers with a finite number of digits. Discuss the concept that Pi is a ratio of two finite integers.

Re: How real are real numbers? (2004)

#27

Post by enugu »

Smaug123 wrote: post #18
enugu wrote: post #17

Earlier quoted context omitted.

I think you (and some other commenters) are responding to something different - potential vs actual infinity - not being able to write down all natural numbers, but we can potentially write any number(with some assumptions like an infinite universe). Whereas the point being made is different and needs some math background which is the work of Cantor. Countable can include all natural numbers that you count until infi…

Nitpick: the infinite countable union of countable well-ordered sets is countable. This statement is immune to the failure of Choice.

As Lang used to say the Axiom Of Choice is obvious, 'I just pick the elements'. Just kidding, thanks for this interesting logical point.

Re: How real are real numbers? (2004)

#29

Post by upquark »

Smaug123 wrote: post #24
upquark wrote: post #21

Earlier quoted context omitted.

Wait, you're agreeing with me that the set in question is ill-defined, but still somehow comparable to other sets? I am not sure I follow. My statement is that you cannot meaningfully talk about "all the numbers that cannot be described with language". If you could, I'd ask you if this set intersected with [0, 1] has a lower bound / 'inf' that's contained in it, and if so, did I just describe that lower bound with la…

Your example is faulty. > I'd ask you if this set intersected with [0, 1] has a lower bound / 'inf' that's contained in it, and if so, did I just describe that lower bound with language? The "indescribable numbers" are dense in [0,1], and so (if the set exists) the inf of the set of indescribable numbers which are between 0 and 1 is 0. Perfectly describable.

My example is incomplete, not faulty. I left it as a question (does the inf belong to the set?). If the answer is yes, we reached a contradiction. If the answer is no, we have to continue further zooming in to this interval (or some other construction along those lines).

See, I claim that this set is ill-defined, so I can't know its properties like whether or not it's dense, open, closed, Borel-measurable, etc. etc.

You have to tell me what its properties are, and I will come up with a concrete proof that the set in question is ill-defined.

EDIT: After I RTFA'd, this is actually the paradox in section 2.3 of the linked article

Re: How real are real numbers? (2004)

#30

Post by btilly »

enugu wrote: post #17
whatshisface wrote: post #3

"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.) I think Borel has confused names with things. The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never…

I think you (and some other commenters) are responding to something different - potential vs actual infinity - not being able to write down all natural numbers, but we can potentially write any number(with some assumptions like an infinite universe). Whereas the point being made is different and needs some math background which is the work of Cantor. Countable can include all natural numbers that you count until infi…

The proof of the reals being uncountable depends on the idea that one can build a number that depends on being able to make an infinite number of choices, each of which depends on the absolute truth or falseness of a statement.

But what happens if we open it up to have statements be true, false, or currently unknown? That is we develop a system of mathematics that could be in principle done inside of a Turing machine?

Then Cantor's diagonalization argument falls apart because of all of those "currently unknown" options. See https://news.ycombinator.com/item?id=13843725 for a previous explanation that I gave of this.

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