I would like to know if the set of undecidable problems (within ZFC or other standard system of axioms) is decidable (in the same sense of decidable). Thanks in advance, and I apologize if the question is too basic.



Is the set of undecidable problems decidable?
No, in fact the situation is even worse than that. The set $T$ of all (Gödel codes for) sentences that are provable from ZFC is computably enumerable; the set $F$ of all sentences that refutable from ZFC is also computably enumerable. These two sets $T$ and $F$ form an *inseparable pair*: $T \cap F = \varnothing$ but there is no computable set $C$ such that $T \subseteq C$ and $F \cap C = \varnothing$.


In other words, there is no finite algorithm that can reliably tell whether a sentence is "not provable" or "not refutable." This is much weaker than asking for an algorithm to tell us whether a sentence is independent or not (in which case we could determine whether it is provable or refutable by waiting until it enters $T$ or $F$). 


This remains true if ZFC is replaced by PA and some still weaker theories.


