[FOM] Fwd: invitation to comment

Vaughan Pratt pratt at cs.stanford.edu
Wed May 25 00:31:07 EDT 2011

Pratt> I'd be very interested to see how T + Con(PA) entails full PA 
where T is less than PA, e.g. PRA.

Tait> That is a misunderstanding of what I wrote. It is Gentzen's 
finitist (or lets say quasi-finitist)
> argument that induction up to epsilon_0 is valid of which I am claiming that, spelled out, requires
> all of PA. That surely should not surprise you.

What does it mean for the validity of Gentzen's argument to "require all 
of PA?"  My impression was the opposite, that PA was a much richer 
system than that embodied in for example Lev Beklemishev's strikingly 
simple worm process as per my previous message.  I don't see ε_0 as 
being any more than what can be found in that process, whereas I see 
"full PA" as embracing the whole arithmetic hierarchy.  In particular 
the game-theoretic machinery of alternation is in full play (so to 
speak) in PA, but where can that whole arena be found in LB's simple 
mechanism, or in ω U ω^ω U ω^ω^ω U ...?

Vaughan Pratt

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