FOM: The logical, the set-theoretical, and the mathematical
Joe Shipman
shipman at savera.com
Mon Sep 11 17:54:04 EDT 2000
In answer to Friedman, Davis, Jones, and Mossakowski:
My position is as follows. Comprehension axioms are "logical". I agree
with Harvey that Set Theory is a branch of mathematics, and therefore
regard the axioms of set theory as "mathematical", though much closer to
being purely "logical" than axioms about other mathematical structures.
HOL (using a deductive calculus with a Comprehension rule as in
Manzano's book) is indeed enough to develop "Ordinary Mathematics".
(Mossakowski insists on an axiom of Infinity as well, but Jones argues
that this axiom is purely ontological and is only necessary in a
metaphysical sense. However, I am willing to accept a modified
logicist thesis that (ordinary) mathematics = logic plus Infinity.)
The Axiom of Choice has a special status. It is not necessary for the
development of number theory, but is certainly an essential part of
ordinary mathematical practice for analysis and algebra. Various forms
of AC are more or less "logical" in flavor but I want to research these
some more before saying whether I think it can properly be considered a
"logical" axiom.
Harvey's insistence that ZFC captures the intuition about sets that
mathematicians had all along (presumably from Cantor c. 1880 to 1930)
seems ahistorical. There were several unsuccessful or incomplete
attempts to formalize sets and classes, and Type Theory must still have
been considered an important alternative (to set theory) foundation of
mathematics as late as 1930 (otherwise, why didn't Godel refer to
Zermelo's system rather than Russell's in the title of his
"Incompleteness" paper)?
-- Joe Shipman
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