On Wednesday 03 May 2006 05:16, dasimov@earthlink.net wrote:
3. I'm not fond of the Axiom of Infinity, either, and it would be very nice if it could be inferred from more natural axioms.
Then you want a different version of set theory. How about NF? The only axioms you need are extensionality and stratified comprehension. (Stratified comprehension is the axiom scheme that gives you every instance of { x : P(x) } in which every variable in P can be assigned an integer "level" so that when "x in y" occurs level(y) = level(x)+1 and where "x = y" occurs level(y) = level(x). Equivalently, take formulae from an old-fashioned "type theory" and erase the types.) In NF, there's a universal set and all the Boolean algebra operations work; an axiom of infinity isn't too hard to prove. You might expect it to fall foul of a Cantorian/Russellian diagonal argument, but it doesn't seem to. (Straightforward attempts founder on the fact that the formula you'd use to define the "diagonalized" set isn't stratified.) So far as anyone knows, NF is consistent. It's also finitely axiomatizable, if you happen to care. On the other hand, the axiom of choice is provably false in NF, which most people find a bit odd. -- g