I will bet $10,000 against Z being found inconsistent within the next 20 years, with odds of 1000:1 against. That is, I'm saying that Z inconsistent is extremely unlikely, and if you think you can make money at any crazy odds on this bet you're the one who's crazy. If you win you get my $10,000, if I win I get only your $10.
These amounts are in US dollars for 2012. We agree that our respective estates will honor the debt.
(I picked the amounts assuming roughly 5-fold inflation over that period.) [I inferred this from the 1.57x inflation increase during 1952-1972 vs. the 3.35x increase during 1972-1992. As it turned out the increase from 1991 to 2011 was only 1.65x, making the point that naive extrapolation of economic trends is unreliable. In hindsight the bets should have been scaled down by a factor of 3, or 5 for rounder numbers.]
Let me know if you wish to accept either bet. I will keep your name private if requested, but not the number of people accepting each bet.
These are meaningful bets. It would be meaningless to propose a bet about the propositional calculus. That can only become inconsistent when all the lights go out.
By "within the next 20 years" I mean by Sept. 19, 2012. Both bets will be payable on Sept. 22, 2012. Alternatively the loser may pay a geometrically smaller amount earlier, namely 1/5 the owed amount in 1992, 8.38% more each year later.
By "found" I mean its ordinary mathematical usage of having a proof that has convinced a majority of reputable mathematicians. This could not happen for either of these bets unless either the proof were derivable in Z or reputations didn't mean as much in 2012.
If there are sufficiently many takers for either bet I will stop taking bets thereafter. (There is no risk of this happening so far.) I currently have no plans to scale down the amounts of the bets; in the unlikely event this changes I will announce it on sci.math. The most likely change would be $100->$10 and $10,000->$1,000 in bet 1. Pleast let me know by email (pratt at cs.stanford.edu) if you would be interested at that rate but not the current rate.
PROBLEM SPECIFICATION
Let me make the problem underlying the bet very concrete. Here are two easily understandable statements.
A. There exists an NxN 0-1 matrix, N the natural numbers, such that every axiom of ZF is true when the variables in the axiom are interpreted as ranging over N and the membership predicate symbol is interpreted so that i is a member of j iff E[i,j] = 1. (So no need to think about uncountable models, or models containing "sets". On the other hand E is too big to write out explicitly all at once, and quantifiers will take forever to evaluate if done naively!)
B. An ideal computer search for an inconsistency in ZF is guaranteed to succeed in a finite time. (Ideal = correct code and no time or space limits. Bugs are up to you. Space: well, how big a proof do you think you're looking for? Time would seem to be your main obstacle.)
B is recognizable as asserting the inconsistency of ZF. By completeness of first-order logic, A is asserting the consistency of ZF, i.e. A = not-B.
Bet 1 is about A vs. B, bet 2 is the same with Z in place of ZF. Z is ZF less F.
We know by Goedel incompleteness that A is not provable in ZF (ditto with Z for ZF everywhere). (On the other hand if either instance of B is provable it is provable in a lot less than Z.) Hence if you plan on trying to prove A you will need some insight about the raw mechanics of A not provable in ZF. As you can see those mechanics are very simple, apart from the big mess of formulas asserting ZF, so you have as good a chance as any mathematician at finding such an insight, possibly more if you don't understand what the axioms mean.
You will find the axioms of ZF explained quite readably as well as formally in Shoenfield's article, pp. 321-341 of Barwise's "Handbook of Mathematical Logic", North Holland, and only formally (!) as Axioms 1-6 in Takeuti and Zaring, "Introduction to Axiomatic Set Theory" Springer-Verlag (my formal statement of F is their Axiom 5, Shoenfield's version of F is oversimplified).
I would of course appreciate hearing about any substantive discrepancies between these two axiomatizations.
This is pretty much the complete specification of the problem as I understand it.