FOM: "Does Mathematics Need New Axioms?"
JSHIPMAN@bloomberg.net
JSHIPMAN at bloomberg.net
Sun Nov 16 01:14:50 EST 1997
I wish I could read Feferman but I don't have software to read a postscript file
--on Monday I'll take it over to someone who can. Steve says MOST (if not all)
scientifically applicable mathematics OUGHT TO BE formalizable in PA where
Franzen says Feferman claims ALL s.a.m. IS formalizable in PA. These issues
are treated at length in my two papers ("Cardinal Conditions for Strong Fubini
Theorems",TAMS 10/90 and "Aspects of Computability in Physics",PhysComp'92)
which are alas unavailable electronically. If you take a physical theory's
model of reality seriously you get way beyond PA very quickly and even if you
try to replace the functional analysis, etc., with "constructive" equivalents
it doesn't necessarily help. In the case of Pitowsky and Gudder's reformulation
of Quantum Mechanics I showed ZFC doesn't establish the existence of the real
functions they needed (they assumed CH but a real-valued measurable cardinal
would also suffice). If you DON'T take the model of reality seriously and
regard the theory simply as an algorithm for predicting results, OF COURSE an
algorithm is formalizable in PA--but maybe not a proof it converges!-Joe Shipman
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