[atmanthedog]

Unfortunately, math doesn't really permit this type of truth: if your axioms are strong enough to prove general statements about arithmetic, there is no effective procedure to determine whether an arbitrary proof follows from those axioms.

[smadge]

Did you mean to write “there is no effective procedure to determine whether an arbitrary formula follows from those axioms?”

A proof is exactly how we demonstrate that a formula follows from the axioms.

[rocqua]

Still, given a set of axioms, statements will fall into one of three categories. 1) Provably True, 2) Provably False, 3) Undecideable

Claims that a statement is in category 1 are fully verifiable (by providing the proof). The same goes with claims that a statement is in category 2.