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What happens if our mathematical system is proven to be inconsistent? (lemmy.ml)

submitted 11 hours ago by HiddenLayer555@lemmy.ml to c/asklemmy@lemmy.ml

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From my "watched a YouTube video" understanding of Gödel's Incompleteness Theorem, a consistent mathematical system cannot prove its own consistency, and any seemingly consistent system could always have a fatal contradiction that invalidates the whole system, and the only way to know would be to find the contradiction.

So if at some point our current system of math gets proven inconsistent, what happens next? Can we tweak just the inconsistent part and have everything else still be valid or would we be forced to rebuild all of math from basic logic?

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[–] MalReynolds@slrpnk.net 9 points 8 hours ago (1 children)

I'm not going to dive in there at the moment, correct me if I'm wrong (the 'the problem would get fixed up and things would keep going about like before' case I suppose).

To answer OP's question, basically the same thing that happened last time with Russel's paradox, which, to massively simplify, was add a new axiom that says you can't do that and carry on. (Which gave rise to the aforementioned ZFC set theory).

[–] solrize@lemmy.ml 5 points 8 hours ago

Yes, that's Frege's system mentioned in Henry Cohn's post. But that happened in a very naive time compared with today. So it would be more of a surprise if something like that happened again.