[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.