FOM: Infinity

[Neil Tennant]

On Thu, 27 Sep 2001 JoeShipman at aol.com wrote, a propos the distinction
between finite and infinite:

> This distinction has been satisfactorily formalized by the use of the
> strongly equivalent systems Peano Arithmetic (PA) and Finite Set
> Theory (the theory of the hereditarily finite sets, axiomatized by ZF
> with the Axiom of Infinity replaced by its negation).

FST is, presumably, a first-order theory. It has an infinite model. Hence,
by the upward Lowenheim-Skolem-Tarski theorem, it has models of every
infinite cardinality. In an uncountable model of FST, each individual will
be borne the membership relation by at most finitely many other
individuals. But isn't it somewhat unsatisfactory to have a theory of the
finite true in a world in which there are uncountably many finite things,
even if nowhere in such a world is there an even countably infinite thing?
Has the distinction really been satisfactorily formalized as Shipman
claims?

Neil Tennant