If, one fine day, someone found a contradiction in ZFC (or even ZF), what implications would such an event have for mathematicians? Is there currently any backup axiomatic system on par with ZFC that would enable mathematicians to continue pursuing mathematics, should the unthinkable occur?
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There was a quite relevant post at here:
and the original lecture is worth watching.
The way I see it there are two possible outcomes:
In either cases there will be a lot of work to see which fragments of ZFC we can rescue, and how many of the proofs would go through. Note that Russell's paradox, at the time, wasn't considered as a big deal by non-logicians. So while the second option would affect many mathematicians, the first option would be largely ignored by most (or it will be used as a source to kick set theory around).
If at some point we conclude that we cannot rescue a lot of the reasonable proofs (e.g. some inner model theory, forcing, etc.) then it is possible that ZFC will be just dropped and a new basis will be sought.
Of course it is possible that just something about large cardinals would be inconsistent, and then we will be able to draw a lot of conclusions about the universe of sets (for example, all propositions which imply the consistency of this cardinal would be immediately false. If you have found a contradiction in a measurable cardinal then you can prove that AD is inconsistent with ZF). But that's still quite far from showing that ZFC itself is inconsistent.
Note that an inconsistency implies that any theorem and its negation can be proven. The fact such a thing hasn't happened (no one has ever proven the negation of a theorem in mainstream math) means that wherever ZFC fails, it is almost certain that a more restricted set of axioms can be found where most math can still be done.
To see an example of what could happen you have to look no farther than Russell's Paradox. It shows that naive set theory makes no sense, but the example is far removed from the regular sets (with "actual elements") that we all usually use, and axioms where set theory works were found.
protected by Asaf Karagila Jan 4 '14 at 19:21
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