Significance of Gödel's Second Incompleteness Thm?

[The idea]

In an inconsistent system, any statement can be proven. In particular, a statement that amounts to "the system is consistent" can be proven. So the existence of a consistency proof would not actually show that the system is consistent. It could be that we are facing a situation where the system is inconsistent and where we have a proof, within that inconsistent system, of an encoding of "the system is consistent." In other words, if we had a consistency proof, then it wouldn't do us any good anyway. So why should anyone be concerned that we can't produce a consistency proof?

That a statement can be proven within a given system has no significance unless we assume that the system is consistent. So a consistency proof is not of any significance unless we assume that the system is consistent. If we are simply going to assume something anyway, then why bother to make an effort to prove it?

[69dodge]

You're right that no proof, including a proof of consistency, would convince anyone who doesn't already believe that the proof system is consistent. (And, in fact, in light of Goedel's theorem, a proof of consistency should convince anyone that the system is not consistent.) But it's still interesting that a consistent system can't prove the true statement that it is consistent. Such a proof might not be useful, but it still---one would think---ought to be possible. The conclusion is true, after all, so why should it not be provable?

(A recent thread here about Goedel prompted me finally to read his paper. I had read a lot about it, but never actually read the paper itself. I found two translations online, an older one (http://home.ddc.net/ygg/etext/godel/) whose English I prefer but which has numerous typos in the math, and a newer one (http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel-errata.html) whose math is better (though still not perfect) but whose English is worse. Also, the newer one doesn't include the footnotes, some of which are rather important. So I'm currently reading them in parallel, taking the best parts of each. :D)

[drkitten]

Originally posted by The idea
In an inconsistent system, any statement can be proven. In particular, a statement that amounts to "the system is consistent" can be proven. So the existence of a consistency proof would not actually show that the system is consistent. It could be that we are facing a situation where the system is inconsistent and where we have a proof, within that inconsistent system, of an encoding of "the system is consistent." In other words, if we had a consistency proof, then it wouldn't do us any good anyway. So why should anyone be concerned that we can't produce a consistency proof?

That a statement can be proven within a given system has no significance unless we assume that the system is consistent. So a consistency proof is not of any significance unless we assume that the system is consistent. If we are simply going to assume something anyway, then why bother to make an effort to prove it?

Because in inconsistent systems, proofs that depend upon the inconsistency can usually be found out pretty fast. In a system in which anything can be proven, the same proof structure will not only prove X, but will prove related things like not-X with extremely minor changes. For example, the same proofs that yield 1=0 will also yield 2 = 0, 2 = 16, 23 = 408, et cetera.

If we actually had a proof of a formal system's consistency in-hand, it would probably relatively easy to inspect it to determine if the proof hinged on system inconsistency or not.