[FOM] predicative foundations (Harvey Friedman)
mlink@math.bu.edu
mlink at math.bu.edu
Thu Feb 16 16:15:43 EST 2006
Dear Professor Friedman,
Concerning the question of whether Kurt Goedel ever
subscribed to one of the controversial stopping places
for predicativity, I look forward to reading Professor
Weaver's responses to your two questions. In the
meantime one important resource is Goedel's December
1933 lecture to the Mathematical Assocation of America,
which is 1933o in volume III of the Collected Works
(pages 45-53).
On this lecture Professor Charles Parsons in
`Platonism and mathematical intuition in Kurt Goedel's
thought' (Bulletin of Symbolic Logic 1 (1995), 44-74)
writes: `Much of it is devoted to the axiomatization of
set theory and to the point that the principles by
which sets, or axioms about them, are generated
naturally lead to further extensions of any system' to
which they give rise. Parsons continues: `When he turns
to the justification of the axioms, he finds
difficulties: the non-constructive notion of existence,
the application of quantifiers to classes and the
resulting admission of impredicative definitions, and
the axiom of choice'.
Goedel summarizes his position (on page 50 of
1933o): `The result of the preceding discussion is that
our axioms, if interpreted as meaningful statements,
necessarily presuppose a kind of platonism, which
cannot satisfy any critical mind and which does not
even produce the conviction that they are consistent'.
Parsons observes it `is clear that Goedel regards
impredicativity as the most serious of the problems he
cites and notes (following Ramsey) that impredicative
specification of properties of integers is acceptable'
under a certain assumption, namely that, as Goedel states
(same page as above): `the totality of all properties
[of integers] exists somehow independently of our
knowledge and our definitions merely serve to pick out
certain of these previously existing properties'.
This, Parsons writes (same page), is a `major
consideration prompting him [Goedel] to say that
acceptance of the axioms ``presupposes a kind of
platonism'''.
Sincerely yours,
--ml
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