Zero-knowledge proof that 0 = 1 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 zero-knowledge 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.)
 A: 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 zero-knowledge 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.
A: Well you're not going to prove 0=1 in PA, because PA is consistent,
(though not PA-provably 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 Zero-Knowledge
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 
self-evidence 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.
A: I doubt it.  If nothing else, you would have a proof of P=NP as well, and since zero-knowledge proofs depend on the hardness of certain problems, you would "have a proof" that you could not have a zero-knowledge 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
A: 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.
http://en.wikipedia.org/wiki/PCP_theorem
