FOM: Two questions
Mark Steiner
marksa at vms.huji.ac.il
Tue Jan 25 14:40:46 EST 2000
Since one of the purposes of this list is allow nonprofessionals to
learn something about the foundations of mathematics from the
professionals, I'd like to pose two (unrelated) questions:
1. Every "core" mathematician knows the difference in modes of reasoning
between real analysis and complex analysis. Famous mathematicians like
Ulam talk rapturously about a "new world". Yet the various
"hierarchies" of logic that I'm familiar with cannot distinguish real
from complex analysis, given that formally the only difference between
them is that the complex numbers are ordered pairs of reals (or can be
modeled that way). Has any work been done to make these finer
distinctions? To me it seems that the thrust of f.o.m. seems to be to
reduce distinctions, e.g. how much you can do with how little. On the
other hand, I have not read all the postings carefully, and maybe this
question has been addressed already. If not, I think it should be,
particularly if you feel, like Martin Davis, that philosophy (and a
fortiori f.o.m.) should connect with what mathematicians actually do.
2. This is something that has been bothering me since Harvey and others
posted claims that ZFC is a "foundation" for all mathematics, but I was
motivated to ask after finally receiving the George Boolos memorial
issue of Philosophia Mathematica. In his work on Frege, George pointed
out that Frege, among many other things, can be regarded also as a
"core" mathematician, since he proved a nifty theorem as part of the
Grundlagen, namely that every natural number has a successor (to be more
exact, his proof is only sketched in the Grundlagen, and it is incorrect
in the Grundgesetze, but was fixed by Boolos). He does this by
induction, a principle which for Frege does not imply the actual
existence of a unique successor for each natural number, so it's not
strictly circular. But it certainly looks like getting something for
nothing. (And thus, IMHO, it has g.i.i.--it was attacked, for example,
by Poincare as impredicative. I think also Frege needed what Harvey
calls "philosophical acuity" to think of such a "free lunch" idea.) The
induction step goes that if n has a successor, then so does any
successor of n. Using Charles Parsons' notation, Nx means "the number
of x such that", he uses the lemma that Nx(n is a natural number and x
is no greater than n) succeeds n. However, this lemma I think can't be
proved in ZFC. It is notorious that Frege defined Nx(Fx) as [F], the
equivalence class of concepts H such that the extension of H can be
mapped one to one onto the extension of F. And [F] can't be defined in
ZFC. Of course trying to formalize this argument led to Russell's
Paradox, though Boolos and others show that it can be formalized in a
natural way, since the only extensions used in the proof are of the form
[F]. Would this, then, count, as a piece of core mathematics not
formalizable in ZFC? If yes, might there not be others?
More information about the FOM
mailing list