Does there exist a non-trivial Ultrafinitist set theory?
Does there exist a set theory T-which has not yet been proved to be inconsistent-and in which
one can prove the existence of (1) the empty set (2) sets that are singletons and (3) sets which
have non-empty proper subsets. T has no axiom of infinity but-as with Quine's NF-one can prove in
T the existence of a universal set (i.e a set of all sets). However-unlike Quine's NF-the universal
set of T should be finite. One can think of T as being formalized in the classical first order predicate
calculus, using the same language as ZF.
My motive in seeking a set theory such as T is to find out whether there exist set theories that might
be acceptable to an ultrafinitist (as conforming to the principles of that viewpoint), while still
allowing a certain amount of arithmetic to be carried out in them.