FOM: Neo-Fregean reverse mathematics (Hazen)
Richard Heck
heck at fas.harvard.edu
Tue Mar 27 10:30:41 EST 2001
Allen's remarks (that FA is not categorical and that its theorems are of
higher degree than those of PA2) give additional substance, it seems to me,
to a complaint made repeatedly by George Boolos, namely, that HP ("Hume's
Principle") is a lot stronger than it needs to be for the purpose of
deriving arithmetic (PA). I would guess the situation is not as bad with
respect to the weakening of HP that was introduced to resolve some of that
problem, namely, FHP:
Finite (F) v Finite(G) --> [#F = #G iff F~G]
where 'Finite(F)' is some sensible second-order formula saying that F is
finitely instantiated. (See my "Finitude and Frege's Theorem" for the proof
that FHP will get you PA2 and, further, that FHP can be proven from PA2,
the usual Fregean definitions, and (if you're not assuming only numbers are
in the domain of PA2) the additional axiom PAF: Every predecessor of a
natural number is a natural number.)
On the other hand, though, there is some need for care here. PA2 is usually
formulated in such a way that the natural numbers exhaust the domain: It's
only if one does formulate it that way that it is categorical. FA, and
variations on it, are not usually formulated that way: It's not assumed
that only numbers are in the domain. And if one does assume that, then it
is NOT true that FA has models of every infinite cardinality: It will have
a model of size k if, and only if, k = the number of cardinals less than or
equal to k. (My set theory is old and rusty: Does that mean k is a regular
limit?) As for what significance that might have, I'll throw up my hands
with Allen.
Richard
----------------------------------------------------------
Richard Heck
Professor of Philosophy
Harvard University
Research Interests: Philosophy of Mathematics, Logic, Language, and Mind; Frege
E-mail: heck at fas.harvard.edu
Personal Web Site: http://www.people.fas.harvard.edu/~heck
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