> Similarly, it seemed to David Hilbert like it should be possible to have a system of formal logic that all could accept, which would prove itself to be consistent.
Minor nitpick, but I don't think this fully captures the essence of Hilbert's program. If we had a system that could prove itself consistent, that would mean nothing, because the system could be inconsistent, therefore unsound and so its consistency "proof" wouldn't mean anything.
Rather, the goal was to start from an arguably "uncontroversial" base of mathematics (something like primitive recursive arithmetic, which intuitionists might also accept) and use that to bootstrap "infinitary mathematics" of the kind the formalists such as Hilbert were arguing for. The hope was that one could prove the consistency of such more abstract mathematics (PA or ZFC) from the more concrete foundations.
Unfortunately, Gödel's theorems completely demolish that idea. It's not just that a (sufficiently strong) theory can't prove itself consistent, it also can't be proven consistent by a strictly weaker theory such as PRA (because if such a proof existed in the weaker system, it would of course also exist in original system).
Minor nitpick, but I don't think this fully captures the essence of Hilbert's program. If we had a system that could prove itself consistent, that would mean nothing, because the system could be inconsistent, therefore unsound and so its consistency "proof" wouldn't mean anything.
Rather, the goal was to start from an arguably "uncontroversial" base of mathematics (something like primitive recursive arithmetic, which intuitionists might also accept) and use that to bootstrap "infinitary mathematics" of the kind the formalists such as Hilbert were arguing for. The hope was that one could prove the consistency of such more abstract mathematics (PA or ZFC) from the more concrete foundations.
Unfortunately, Gödel's theorems completely demolish that idea. It's not just that a (sufficiently strong) theory can't prove itself consistent, it also can't be proven consistent by a strictly weaker theory such as PRA (because if such a proof existed in the weaker system, it would of course also exist in original system).
Interestingly, there are systems which are neither stronger nor weaker than PA which can prove PA consistent: https://en.m.wikipedia.org/wiki/Gentzen%27s_consistency_proo...