[FOM] Adequacy of FOL+ZFC as foundation
joeshipman at aol.com
Fri May 14 01:52:39 EDT 2010
On an earlier thread, I mentioned two ways that first-order logic with
the ZFC axioms is problematic as a foundation:
1) Grothendieck Universes are necessary for the convenient development
of some modern number-theoretic results
2) NF is not consistent with ZFC so far as we know.
McLarty's paper on Wiles's theorem suggests that a metatheorem is
possible which would eliminate the need for Grothendieck Universes,
although he does not state one. (I can state one, that you don't need
Grothendieck's axiom of a proper class of inaccessibles -- a single
inaccessible limit of inaccessibles suffices for any application to
number theory or indeed for any "Absolute" statements.)
No one responded to my request for examples of mathematics done in NF
as a base theory that were not obviously reinterpretable as arguments
Therefore my two examples do not appear to lead to statements which are
generally regarded as having been proven, but which cannot be seen to
be theorems of ZFC.
Can anyone suggest another example that might show the inadequacy of
FOL+ZFC? I would like to exclude statements that require large
cardinals, because I can regard them as simply being ZFC theorems about
the statement following from a large cardinal.
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