Math went on because it doesn't matter. Nobody cares about incompleteness. If you can prove ZFC is inconsistent, do it and we'll all move to a new system and most of us wouldn't even notice (since nobody references the axioms outside of set theorists and logicians anyway). If you can prove it's incomplete, do it and nobody will care since the culprit will be an arcane theorem far outside the realm of non-logic fields of math.

[–] [email protected] 2 points 1 year ago* (last edited 1 year ago) (1 children)

You wouldn't even notice if some proof is wrong because it relies on an inconsistency that's the issue. And that's before you didn't notice because noone builds anything on axioms but instead uses fragile foundations made of intuition, hand-waving, and mass psychology.

Incomplete is fine that's exactly what constructive maths is doing.

[–] [email protected] 1 points 1 year ago

You wouldn’t even notice if some proof is wrong because it relies on an inconsistency that’s the issue.

You wouldn't notice because there's no realistic chance that any meaningful result in the vast majority of math depends strictly on the particular way in which ZFC is hypothetically inconsistent.

And that’s before you didn’t notice because noone builds anything on axioms but instead uses fragile foundations made of intuition, hand-waving, and mass psychology.

This is a ridiculous attitude. Nobody uses the axioms of ZFC directly because that would be stupid. It's obviously sufficient to know how to do so. There is literally no difference to the vast majority of all math which particular axiomatic formalism you decide to use, because all of those results are trivially translatable between them.