TITLE: Does there exist a non-trivial Ultrafinitist set theory?
QUESTION [8 upvotes]: 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.

REPLY [5 votes]: I have been studying a theory I call Modular Arithmetic (MA). MA has the same axioms as first order Peano Arithmetic (PA) except Ax( ~S(x)=0 ) is replaced with Ex( S(x)=0 ). MA is consistent because it has arbitrarily large finite models base on modular arithmetic. The upward Löwenheim–Skolem theorem proves MA must have infinite models. MA can be made into an ultra-finite theory by adding an axiom like Ax( x=0 or x=S(0) or ... or x=Sn(0) ) where Sn(0) is some finite number of applications of successor to 0 (a numeral). 
Coming up with a set theory based on MA is problematic. It is simple to show MA is omega inconsistent. The predecessor of 0 must be non-standard (not a numeral) in any infinite model of MA. If the predecessor of 0 is standard we can prove the model is finite using induction. This means Ax( ~S(x)=0 ) is true for all standard natural numbers in any infinite model. A set theory based on MA can't be well ordered or well founded, either. (x =< y) <-> Ez( x+z=y ) is trivially true for any x and y. Using S(x) = x U {x} as a definition of successor is inconsistent with the axioms of MA.
In ZF, even the Axiom of Pairing allows the construction of arbitrarily large sets. Assuming we could come up with a set theory for MA, it would have the properties you ask for. One way to do this would be to encode sets as binary expansions of natural numbers. Element x is a member of the set if the x'th bit of the expansion is true. This would allow us to have sets of size log2(n) where n is the size of the universe. We can equate 0 with the empty set. We can define singleton sets for "small" elements of the universe. We could also define sets with subsets. We could do a reasonable amount of arithmetic by choosing n large enough. We could have sets and do math on sets as large as 100 by having a universe of size 2^100.