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The second theorem tells us that we can't prove the consistency of set theory without using an even more powerful system. This doesn't mean that set theory is i
by cchooper 18y ago
The second theorem tells us that we can't prove the consistency of set theory without using an even more powerful system. This doesn't mean that set theory is inconsistent, it just means that we can't prove its consistency in any meaningful way. So yes, Gödel indeed shows that there are no 'ultimate axioms' that contain everything of interest in mathematics.
- mstoehr 18y agoAlthough it does lead to a mathematically uninteresting paradox: if you let A be the axioms of set theory and you add an axiom P which states that A proves x and not x, (i.e. set theory is inconsistent) then A' = A and P is still consistent.