[mcherm]

Consider the set of all real numbers which you can actually specify IN ANY FASHION AT ALL, so that the person writing a paper about it and the person reading the paper are talking about the same number. This includes easy ones like "3" or "Square root of Pi". It includes oddballs like Chaitin's constant -- the probability that a randomly constructed program will not get stuck in an infinite loop (for some particular choice of computer and programming language) -- which is so ornery a number that we cannot (even in THEORY) figure out a single digit of it (other than its being between 0 and 1).

Consider that great big set. It's still countable. The thing that makes "real numbers" bigger than integers, the "extra" that real numbers have must be very, very peculiar.

[GregBuchholz]

>Chaitin's constant -- <snip> -- which is so ornery a number that we cannot (even in THEORY) figure out a single digit of it (other than its being between 0 and 1).

You might be interested in:

https://www.cs.auckland.ac.nz/~cristian/Calude361_370.pdf

"A Chaitin Omega number is the halting probability of a universal Chaitin (self-delimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random(its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly noncomputable. The aim of this paper is to describe a procedure, that combines Java programming and mathematical proofs, to compute the exact values of the first 64 bits of a Chaitin Omega: 0000001000000100000110001000011010001111110010111011101000010000"

[tromp]

Unfortunately, that is not a Chaitin Omega, since the notion of program length in that paper is flawed. In particular, it counts a 0 or a 1 provided as binary data in the program as adding 7 bits of length to the program. Later papers by Calude fix this problem, while reducing the number of computed bits to 43.

[stymaar]

> Consider that great big set. It's still countable.

You didn't prove that statement, and actually Cantor's diagonal arguments [1] proves you wrong:

Consider the set of «all real numbers which you can actually specify IN ANY FASHION AT ALL». If you can count it, you can order them in a certain fashion: n0, n1, n2, n3 … It's easy to specify a number X which is not part of this set (which is in contradiction with the definition of this set, and demonstrates ad absurdum that this set is not countable). To build such X, consider the n-th decimal of the n-th number: if it's zero, the n-th decimal of X is 1, otherwise the n-th decimal is 0. Then by construction, X is different from any number of this set, which mean X is not a part of the set. □

[1]: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

[moefh]

I don't think that argument works.

How do you guarantee that you can compute the n-th decimal of the n-th number in the list? In fact, from what I understand, this paper[1] shows how to specify exactly a number can't be computed like that.

[1] https://arxiv.org/pdf/1003.0480.pdf

[stymaar]

Why do you need to compute the n-th decimal of a number for this proof to work ? let Ω be a Chaitin Omega number it's decimal are non-computable, yet for every x, Ω + x is a perfectly valid number.

Then `floor(Ω.10^(n))-10.floor(Ω.10^n-1)` is also a number (a natural one) and actually it's the n-th decimal of Ω. It cannot be computed, but it still exist no matter what.

[gertef]

The number you are paraphrasing cannot be precisely identified in finite time.

[stymaar]

Oh, you're right. For that I would need to explicit the construction of the sequence n0, n1, … but that's impossible : if I can find such sequence, my proof holds but at the same time such sequence shows that the set is countable => contradiction, hence there is no such sequence.

But well, now that we've proven that such sequence doesn't exist, we've proven that the given set is not countable !

Not the most elegant proof ever, but I think it works.

[stymaar]

Actually it doesn't: what if the given sequence exists but cannot be expressed with words ? (This is exactly the same thing than «a real number that can't ne expressed with words» which is what I'm trying to demonstrate not to exist. I'm going nowhere)

[deepsun]

What about phrases that specify some numbers on Thursdays, and another numbers when Moon is in second quarter?

What about phrases that some people agree specifies a number, while other people think it's another number, and yet another people just aren't sure?

What about phrases that specify one and the same number for a specific person, but once that person reached age of 40, then he/she is not sure anymore?

I just making counter-examples to Jules Richard paradox formulation, showing that it's not properly formalized, but rather expressed in vague language, so cannot be considered strict mathematics.

[E6300]

> What about phrases that specify some numbers on Thursdays, and another numbers when Moon is in second quarter?

Would you mind producing such a phrase?

> What about phrases that some people agree specifies a number, while other people think it's another number, and yet another people just aren't sure?

> What about phrases that specify one and the same number for a specific person, but once that person reached age of 40, then he/she is not sure anymore?

You may as well ask whether someone understanding a theorem has any bearing on its truth value. A well-formed sentence in a formal language has one and only one meaning regardless of whether a specific person understands it. That's the whole point of using a formal language. If meaning is actually somehow bound to its interpretation then communication of even the simplest of mathematics is totally impossible.

[deepsun]

> Would you mind producing such a phrase? Sure: "Current quarter of the Moon". Or "How Giants scored last season". Or "One, if P=NP, zero otherwise".

Well, I see your point, if we limit our phrases to only formal language, then you're right.

[E6300]

> "Current quarter of the Moon". Or "How Giants scored last season"

These sentences describe functions, not numbers.

> "One, if P=NP, zero otherwise".

This value is a constant. It doesn't change through time. We simply don't know what it is.

[deepsun]

> These sentences describe functions, not numbers.

Yes, and if we think deeper about it, we'll find out that in mathematics we very often use functions in place for numbers, like it was the same thing. Simple example: √2. Even it form suggests that we apply function "square root" to number 2. Less obvious example: π. It's also a relation between diameter and circumference.

If we go even deeper, trying to understand what, for example, number "2" means, we'll come up to their definition through functions (or classes) of equivalence of sets cardinalities (cardinal numbers), or order (ordinal numbers), where sets are defined using ZFC, for example.

I met this discrepancy while naively trying to program mathematical logic in college. In mathematics you just "take" a number, while in programming you "create" or "instantiate" a number, and it has a ton of consequences.

>> "One, if P=NP, zero otherwise".

> This value is a constant. It doesn't change through time. We simply don't know what it is.

Not necessarily a constant, as it's not proven to be a constant yet. As an example of how it may turn out to be non-constant, check out Continuum hypothesis proof (https://en.wikipedia.org/wiki/Continuum_hypothesis) : solution is independent of our axiom set. In other words -- we can state that ℵ1=c, or we can say ℵ1!=c, and both statements will not contradict anything else. We'll just get two different models, with one more axiom each.

[E6300]

> Yes, and if we think deeper about it, we'll find out that in mathematics we very often use functions in place for numbers, like it was the same thing. Simple example: √2

√ is a function. √2 is the result of applying √ to 2, which is a number. Since "how Giants scored last season" is a function, it can be applied to a value such as "2017-04-17", therefore "how Giants scored last season, and today is 2017-04-17" would be a number.

I don't follow your P=NP argument. Either P=NP or P!=NP. They can't both be simultaneously true. I also don't follow how the continuum hypothesis is relevant. = is not comparing the cardinalities of P and NP, it's asking whether they're the same set (i.e. all elements of P are in NP and vice versa). Again, either they're the same set or they're not. Even if there was a cardinality between that of N and that of R, I don't see how that would change how we compare sets for identity.