[Math] Does there exist a non-trivial Ultrafinitist set theory

lo.logicmathematical-philosophynt.number-theoryset-theoryultrafinitism

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.

Best Answer

Perhaps I have misunderstood your requirements, but it seems that the following trivial theory has all your desired properties. Namely, let $T$ be the theory asserting

  • There are exactly three distinct objects: $E$, $P$ and $V$.
  • $E$ has no members.
  • $P$ has only $E$ as a member.
  • Everything is a member of $V$, including $V$.

This theory is clearly consistent, since we can write down a finite model, with three elements. In fact, this is the only model of $T$. But meanwhile, it has all your properties, since it asserts that $E$ is empty and that $P$ is a singleton and that $V$ has a nonempty proper subset, namely, $P$. Finally, $V$ is finite, since it has exactly three elements.

(One quibble, you said in (2) that you wanted "sets" and not just a set that was a singleton. In this case, please add another set $Q$ to the theory that has only $P$ as a member, and also make $Q$ an element of $V$.)

I suspect that you may have in mind that the theory should also include additional unstated set-theoretic principles.

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