On Mon, 31 Jul 2006 21:46:52 +0000, Jim Heckman wrote:
> I'd be interested to know which ZF axioms your "imagine[d] reasonable > people" don't believe.
For some people it's not a case of "believing" or "not believing" ZF axioms. It's rather a matter of not believing in a single objective world of sets.
Compare the situation with Euclidean/non-Euclidean geometry: we don't have to declare that we "believe" or "don't believe" the parallel postulate. (Such a declaration would only mean anything if we were referring to some objective world, e.g. our physical universe.) You simply study whatever geometrical system fits your purpose. Similarly, you can study whatever set-theoretic system fits your purpose.
For example, if you're writing about combinatorics you might declare "in this paper, all sets will be assumed finite". You might only be doing this in order to save having to write the word "finite" over and over again. On the other hand, the chances are you'd be doing various operations on your finite sets (forming products, taking power-sets, etc), and that would depend on the fact/supposition that the world of finite sets admits such operations - obeys some of the ZF axioms, if you like. In terms of belief, it could be said that you've temporarily suspended your belief in the axiom of infinity. But I don't think "belief" is a good way to look at it.
One variant of your question is: in what ways could you modify the ZF axioms and still reasonably call them axioms for "sets"? Obviously this is a fuzzy question, but it's not so fuzzy as to be meaningless. E.g. I suppose most people would agree that if you add the Axiom of Choice then you could still reasonably say that the result (i.e. ZFC) is a system of axioms for some kind of set theory, but no one would agree that if you threw out all the ZF axioms and replaced them with axioms for the complex numbers then those could be called axioms for a set theory.