[FOM] Could PA be inconsistent?

Charles Parsons parsons2 at fas.harvard.edu
Fri May 14 10:41:17 EDT 2004

At 4:18 PM -0700 5/13/04, Martin Davis wrote:
>There are now two threads going on FOM contemplating this 
>possibility. No one has mentioned that we possess proofs that PA is 
>consistent. These proofs can be carried out in quantifier-free 
>primitive recursive arithmetic with induction permitted on a 
>primitive recursive well-ordering of the natural numbers of order 
>type epsilon-0. These proofs (Gentzen, Ackermann, G\"odel) have gone 
>unchallenged for decades.

One could view these proofs as relative consistency proofs, in a much 
weaker system, of PA relative to quantifier-free primitive recursive 
arithmetic with induction up to any specific point less than 

Someone who thinks it a real possibility that PA is inconsistent no 
doubt will try to argue that it is a real possibility that that 
system is inconsistent. I think that would be harder to argue, but 
not being a player in this game, I don't know how much harder.

Charles Parsons

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