FOM: Infinity JoeShipman at
Thu Sep 27 16:24:13 EDT 2001

>>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?<<

I was not writing about "the distinction between finite and infinite", but rather about the distinction "between finite mathematics and mathematics that uses infinity".

Peano Arithmetic (or Finite Set Theory) does a good job of capturing finitary *reasoning*, even if the philosophical role of finite vs. infinite *sets* is not perfectly clarified.

By the way, are there any nice finitely axiomatizable systems which stand in the same relation to Peano Arithmetic or to Finite Set Theory as Godel-Bernays set theory stands to ZFC? (that is, they may introduce new predicates but are conservative extensions with respect to sentences which don't use the new predicates).

-- Joe Shipman

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