So the halting problem basically states that there cannot exist any finite length algorithm for automatically verifying if other finite length algorithms terminate.
But suppose I start listing out all the possible programs. I am allowed to list them as they form a countable set involving finite length permutations of a countable number of symbols. We can label each program $X_0$, $X_1$ ... where $X_n$ denotes the n'th program that is listed by some scheme of selecting and permuting symbols.
Obviously an algorithm cannot exist that is successfully able to verify if ALL of the programs halt or do not. But does there exist any program $X_i$ for which it is simply not possible to verify if it will halt or not?
My analysis of it so far:
Assume we discover a program X that isn't verifiable. In other words there is a proof that there does not exist any algorithm A to determine if X terminates or does not. Then simply running X cannot result in us finding out if it terminates.
Thus X does not terminate in any finite amount of time. Thus X doesn't halt. Therefore we have concluded that X is verifiable. Meaning there doesn't exist an undecidable program X for which there exists a proof that X cannot be verified by an Algorithm A in a finite amount of time.
This however DOES NOT Mean every program X either can be proven to halt or not. Rather it states the programs X that are undecidable themselves do not have a proof of this fact. Here is the catch, if there is a proof that a program X cannot be proven to be undecidable then it must be the case that X is decidable so if X is undecidable then even this type of a proof cannot exist.
We conclude that if X is undecidable, then there doesn't exist a proof of X is undecidable, nor a proof that there doesn't exist a proof of X being undecidable.
Here is another form of the question, if I arbitrarily continue this chain of analysis I believe I will always conclude that X doesn't exist. That is if $$\exists \text{Proof} (\nexists \text{Proof of} \left( \nexists \text{Proof of} \left( \nexists \text{Proof of} \left( ... \left(\text{Program X is undecidable} \right) \right) \right) \right) $$ Then,
X is decidable.
Whats a hint for beginning to prove this result. And if such a proof exists, what philosophical implications does that have?