[FOM] Fwd: invitation to comment
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 ...?
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