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Stuart_Armstrong comments on The Last Number - Less Wrong
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Look up: http://en.wikipedia.org/wiki/Self-verifying_theories
"for instance there are self-verifying systems capable of proving the consistency of Peano arithmetic."
Interesting. Why would one want a theory that can prove its own consistency? This doesn't really tell us anything, because inconsistent theories can prove any statement, including their own consistency.
I don't agree that it doesn't tell us anything ... an inconsistent theory can prove all statements, yes, but not all with proofs shorter than its shortest proof of a contradiction. That is, if Peano arithmetic has a trillion-line proof that 3.2 is an integer, then it can prove anything in about a trillion and two lines... but it can't prove everything as easily as say (1+1+1)(1+1)=(1+1+1+1+1+1). It might be something special when a theory can prove its own consistency elegantly, sort of the way a human can have non-zero credence that s/he is usually rational.
I'm not sure I know what a proof of consistency is, except that I wouldn't want an inconsistent theory to be capable of one.
An inconsistent theory can prove anything - including its own consistency.
I'm familiar with "anything statement can be derived from an inconsistent theory" but I really am confused by how any such derivation could be a proof of consistency. If proofs of consistency are possible for inconsistent theories then how exactly are they proofs of consistency?
It's a "proof" in that it follows the formal rules of the proof system. You can "prove" anything if your rules are sufficiently ridiculous, but that doesn't mean the proof actually means anything.
Thanks.
If I tell the truth, I cannot say: "I lie".
But if I lie, I can say: "I tell the truth".
So, a theory's proving its own consistency is strong Bayesian evidence that it's inconsistent ;).
If that's all you know about the theory, I'd say yes - but not "strong" evidence.
I probably should have given more than just a winkie to indicate that I was joking.
Actually, it is a serious point. If you choose thories at random, according to some universal prior, then a lot of them are going to be inconsistent. And most of the theories that can quickly prove their own consistency are the inconsistent ones. So this does provide some information (depending on how the consistency proof was arrived at, of course).
That was pretty much what I was getting at. But since I'm not in a position to quantify how strong the evidence is, I took the cheap route of making it a joke :).
Huh, did not know! Unfortunately they don't imply the results of Peano arithmetic, but that would be asking for too much, heheh.