[drdeca]

> Since you never can finish counting

Not what countable means. The meaning meant is the one about having a map from the set to the integers, where no two things of the set map to the same integer.

The correspondence is meant as either a full thing, or at least a well defined specification which could be applied to any of the things, if you want to get philosophical about ontology or something.

Also, the specification can't depend on things like, what reals have been given as input "previously".

Say that in this game, you have two obligations:

1: for any n,m I give, you must tell me what the mth binary digit of the real number associated with n is, in the mapping you are considering (or, if there is no real number associated with that natural number.).

2: For any finite collection of the binary digits of the real number I am choosing, you must tell me the first natural number such that the corresponding real number matches all the digits I specified.

With these rules, I can choose my real number (specifying more and more digits) such that for any natural number, I will be able to show that my real number doesn't correspond to that natural number, or any lower one.

Therefore, there is no natural number that corresponds to my number in the matching system you are providing.

(To win, I repeat this procedure:

Ask what the first natural not ruled out already by my specification of my number is. (Call that n)

Ask what the nth digits of the real associated with n is.

Inform you that the nth digit of my number turns out to have the other value for its nth digit, so no natural less than or equal to n corresponds to my number.

Repeat.

This will only ever tell you more about my number, and will not result in me changing my mind about any of the digits of my number, yet there is no natural number which won't ever be ruled out as potentially corresponding to my number.

Therefore, no natural corresponds to my number in your system.

So I win.)

[hzhou321]

Why should you get to choose your real number -- specifying more and more digits -- implies an un-exhaustive process, while I am not allowed to do the same? An unfair game will have unfair winners, how would it mean anything? If we both are allowed an un-exhaustive process of specifying what we have, this goes back to the counting game. As we never can finish, how does it make countable (or not)?

[drdeca]

The reason that the real number being defined can be defined in terms of the function is because, that is the situation being considered in the statement.

For any way of doing the mapping, there is a real number that the mapping misses. Alternative statement: "there is no such mapping that doesn't miss any of the reals".

This is shown because, given a mapping, I can find a real that the mapping misses.

When one says "for all x, there exists a y such that P(x,y)", the y is allowed to depend on the x.

That's what this is.

Why wouldn't your objection apply to the proof that the halting problem is uncomputable ? The program that the halting checker can't check is defined in terms of the halting checker. Why is that allowed? Because that is what the statement is saying. For any purported halting checker, there exists a program it doesn't decide the halting of.

Similarly here, for any purported bijection between the integers and the reals, there is a real that the purported bijection misses.

I don't know if you are using the word "countable" in the standard way, so I don't know what you mean by that last sentence.

[hzhou321]

Given any finitely described real numbers, there will be a finitely described mapping map it to a unique natural number. But if you use real numbers that can never finish describing, we have to use a mapping that cannot be finished in describing. That is not the same as "there is no such mapping that doesn't miss any of the reals". If all mappings that cannot be finitely described are excluded, what will be the reason? And why that reason cannot be applied to exclude some real numbers (that cannot be finitely described) as numbers? I understand that is what it is, but being what it is seems meaningless.

The general halting problem is uncomputable. It can only be simulated. However, any program is still assumed to be finitely described. Any discussion in finite domain (including arbitrarily big finite) cannot lead to conclusions or insight toward infinite.

> I don't know if you are using the word "countable" in the standard way, so I don't know what you mean by that last sentence.

In the context of infinity, words such as countable, bigger, order, etc. all lose its standard meaning. We don't really know what it means if we don't really know what infinity is. Mathematicians simply made a definition to countable here -- a finitely described mapping -- that is fine on its own, but completely useless. Since we can't draw any parallels from infinity to finite (including arbitrarily big), we can't really relate any definitions over the infinity to "standard" meaning of those words to the domain of finite.