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Math as we know it, due to Gödel's Incompleteness Theorems, is not fully consistent - we can't even prove everything that is true with our current mathematical
by DecayingOrganic 3y ago
Math as we know it, due to Gödel's Incompleteness Theorems, is not fully consistent - we can't even prove everything that is true with our current mathematical framework. This means that our understanding of math is indeed limited, not just by our intellectual capabilities, but by the very structure of the math we currently use also.
This makes me wonder, will we be able to develop new mathematical frameworks that bypass these issues? And if so, what will they look like?
- kbrkbr 3y ago> is not fully consistent Not being able to prove the consistency of a system within the system does not entail that the system is inconsistent (or not fully consistent).
- pixl97 3y agoI think it means we could only ever prove it is inconsistent, and never prove that it is consistent.
- tux3 3y agoIf we can only prove that an inconsistent system is inconsistent, and we can never prove our (presumably) consistent system is consistent, then it is incomplete. We can't say it's inconsistent just because we can't prove otherwise. We can say it may be incomplete (or inconsistent, and we haven't noticed yet)
- zmgsabst 3y agoThe insight is that mathematics is not complete — which is the property that a system can prove every true theorem. Consistency is there as a technical detail: an inconsistent system can prove every true theorem, by virtue of being able to prove every theorem.