FOM: Ontology in Logic and Mathematics

[Joe Shipman]

Roger Jones wrote:

>>my view is that neither logic nor mathematics need or can make any
absolute ontological commitments, and I doubt that a worthwhile
distinction between logic and mathematics can be founded on ontology.<<

This seems correct, if you are talking about "absolute" ontological
commitments.  But the kind of ontological commitments made by
mathematics are different in kind.  Consider two specific theorems:

1) For all k, l, m there exists n so large that, if you color the
k-element subsets of {1,...,n} with l colors, then there will be a
subset X of cardinality at least m all of whose k-elements subsets have
the same color, and such that the cardinality of X is greater than the
smallest element of X. (Paris-Harrington)

This is  provable in ZFC, but any such proof will use the Axiom of
Infinity.  Does this mean you must be ontologically committed to the
existence of an infinite set in order to accept this theorem as true?

2) The set of finite trees is well-quasi-ordered under the relation of
embeddability (A embeds in B if you can map vertices of A into vertices
of B and edges of A into disjoint paths of B preserving incidence).
(Kruskal, metamathematical analysis by Friedman; equivalent versions
exist which talk only about finite sets.)

This is provable in ZFC, but any such proof will involve both the Axiom
of Infinity and the Power Set axiom.  Does this mean you must be
ontologically committed to the existence of an uncountable set in order
to accept this theorem as true?

-- Joe Shipman