[FoM] consistency of PA?
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
Fri May 7 04:11:45 EDT 2004
Karlis Podnieks writes (commenting on Ford and Holmes):
>>"Must" PA be inconsistent? My idee fixe:
>>If ZFC proves that Goldbach's Conjecture is consistent with PA, then
>>ZFC proves [that] Goldbach's Conjecture [is true]. Is this a "normal"
>>phenomenon, or something like Michelson-Morley? Couldn't it be
>>another starting point for an inconsistency proof?
It does have a whiff of Michelson-Morley about it (somehow the
cards are stacked so you can only get one of the apriori possible
answsers!), but I think it's a "normal" phenomenon. Goldbach (like
the example we all used to use, Fermat) is a Pi-1 statement, so if it
were false it would be refutable in Robinson's Q (or whatever your
favorite Sigma-1 complete but otherwise minimal arithmetic is).
Since the argument to "If it were false, it would be refutable" is a
very elementary bit of metamathematics, available in virtually any
system you care to do metamathematics in, any reasonable system that
can prove
*Goldbach is consistent with PA,
since this amounts to
**Goldbach isn't refutable in PA,
will have to prove
***Goldbach.
Statements of other logical forms CAN be proven consistent with PA
without being proven true: given the 2nd incompleteness theorem, ZFC
(or anything else that can prove the consistency of PA) will prove
that "Not-Con(PA)" (which is Sigma-1) is consistent with PA, but
(many people still hope that) ZFC doesn't prove "Not-Con(PA)" is true.
--
Allen Hazen
Philosophy Department
University of Melbourne
Maker of Obvious Remarks
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