Suppose one day I came up with a proof that 0 = 1 in some formal system such as PA or ZFC that cannot prove its own consistency (unless it is inconsistent). Would it be possible to have a zeroknowledge proof of this? In other words, would it be possible for me to convince you with high probability that I had derived such a proof without (feasibly) revealing the proof of the contradiction? (I haven't by the way...found such a contradiction.)

In this setting, the adversary seeks to find a deduction $\phi_0, \dots, \phi_n$ of $P \wedge \neg P$ quickly. If ZFC, for example, is inconsistent, there exists such a deduction and hence there exists a (constant time) adversary, which simply publishes $\phi$. In order to have a zeroknowledge proof problem, one needs a family of problems, for which the adversary's task becomes increasingly hard as $n \rightarrow \infty$. With just one theory, such as ZFC, this does not happen. 


Well you're not going to prove 0=1 in PA, because PA is consistent, (though not PAprovably so), following Gentzen. But I digress. If you proved 0=1 in, say, ZFC, that would simply mean that ZFC was inconsistent  that the entities it purported to describe had no reasonable interpretation and that logical conclusions derived from the axiom had, in general, no bearing on the world. In particular, it would be irrelevant that you had proved P = NP. But I still digress. My main point: your 0=1 proof is a purely combinatorial object  a symbol sequence that satisfies syntactic constraints that can be checked in polynomial time. The standard ZeroKnowledge Proof technology would apply to this proof just as to any other. The cataclysmic semantics of the proof's conclusion would simply be irrelevant. Surely if ZFC turns out inconsistent, much of set theory could still be saved by suitably weakening say, the particular axiom whose selfevidence turned out illusory. (Consensus in the short term concerning which axiom to give up might turn out difficult to achieve). At the end of the day, the offending axiom would simply seem overambitious, just as the occasional large cardinal axiom turns out to be a turkey, roadkill on the transfinite superhighway if you will. Most of classical mathematics will still go through intact, and the theory of finite sets, PA essentially, already strong enough to articulate the P=NP conjecture, will remain consistent. 


I doubt it. If nothing else, you would have a proof of P=NP as well, and since zeroknowledge proofs depend on the hardness of certain problems, you would "have a proof" that you could not have a zeroknowledge proof. I suggest rephrasing the question so that it is less likely to be closed. Perhaps something like "Has anyone considered the impact of inconsistent theories on zero knowledge proofs (and published their considerations)?" Gerhard "Ask Me About System Design" Paseman, 2010.12.09 


The [[PCP theorem]] says you can give such an argument for any theorem you have a proof of, not just 0=1. Hmm, maybe it relies on consistency though. 

