[JohnKemeny]

Okay, I see what you mean.

It really hinges on what is meant by "this question", i.e., Given the code of a computer program, can you tell whether it will eventually stop or run forever?

If what's given to you is the code and its input, then I think the statement's correct.

However, if the input is assumed to be either the code itself or any other fixed thing, then I agree with you.

[jakelazaroff]

I don't see how what's given to you changes things? If there's ambiguity, I think it's in whether the question is actually the halting problem:

If it is the halting problem — less ambiguously written as "given the code of a computer program and its input, will it run forever?" — then the statement is incorrect: there is a method that returns the correct result for every possible program and input.

If it is about proving whether a machine halts — not getting it right by chance, but showing that a particular machine halts or runs forever — then the statement is correct, for any set of axioms there are Turing machines that cannot be proven to halt or run forever using those axioms.