[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
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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