From: John Rickard Subject: Re: mathematics based on finite sets Date: 02 Oct 2000 18:51:14 +0100 (BST) Newsgroups: sci.math Summary: [missing] Peter Percival wrote: : ZF without the axiom of infinity is equivalent to PA isn't it? I think it is ZF with the axiom of infinity replaced with its negation (rather than simply removed) that is equivalent to PA. I don't know the details, but a model for this modified ZF is the hereditarily finite sets (sets that are finite, all of whose elements are finite, all of the elements of whose elements are finite, etc.); these can be identified with the natural numbers (including 0), by defining 0 = {}, 1 = {0}, 2 = {1}, 3 = {0,1}, 4 = {2}, ..., where m is an element of n iff bit m of the binary representation of n is a one -- I'd guess that the proof of equivalence uses this identification. (You have to choose the right formulation of ZF: I think some like to omit the empty set axiom, since, given the axiom of separation, it follows from the existence of at least one set, which is guaranteed by the axiom of infinity: that's no good in this context, unless you're using a formulation of first-order logic that does not allow an empty universe.) -- John Rickard