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> Peano arithmetic remains consistent. This isn't quite the conclusion here. Peano arithmetic's consistency cannot be proven within its own limits, so it remai
by 0xBABAD00C 5y ago
> Peano arithmetic remains consistent.
This isn't quite the conclusion here. Peano arithmetic's consistency cannot be proven within its own limits, so it remains a hope/intuitive belief, but not a fact. Closest we've gotten, to my knowledge, is that there have been consistency proofs within other axiomatic systems:
https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof
- dllthomas 5y agoI mean, Gödel told us that we can't have a system that proves its own consistency while being consistent, so we shouldn't expect to get closer than "we have proofs in other systems and also the axioms seem so dang simple and obvious and also people have been banging on them and haven't found any inconsistencies..." But moreover, if Gödel's proof had gone the other way I'm not sure the situation is all that much changed. If I hold in my hand a proof of the consistency of a system of axioms within that system of axioms then... either the system is consistent or, by being inconsistent, could prove anything including its own consistency.