[FOM] inconsistency of P and extreme formalism

[Richard Heck]

On 10/11/2011 02:28 AM, aa at post.tau.ac.il wrote:
> These systems can "do syntax" only via *coding* into the ... natural
> numbrs (which we pretend, recall, to doubt/not to understand...).
>
So far as I know, no one's claiming not to understand the natural
numbers. They're claiming not to believe full induction. One might think
(as Burgess somewhere suggests) that the axioms of Q should be thought
of as the axioms of a general theory of cardinality; various forms of
induction can then be seen as attempts to characterize the notion of
*finite* number. No one who thinks that way is claiming that there
aren't any natural numbers though, in a sense, they may be claiming not
fully to understand what a "finite" number is. That is very different.
Why can't one think the whole truth about the natural numbers is given
by, say, PRA? or I\Delta_0 + EXP? or anything else you like? I'm not
saying the view is true, but it hardly seems unintelligible.

Besides which, it's really only conventional here to do syntax via
coding. I could have answered in terms of primitive theories of syntax,
such as Grzegorczyk's theory TC, which is mutually interpretable with Q
(see his "Undecidability Without Arithmetization" and Visser's "Growing
Commas"), and then talked about stronger syntactic theories that are
mutually interpretable with things like I\Delta_0 + \omega_1, which can
of course be interpreted in TC.

Personally, I prefer to think in terms of such theories. Coding, in my
view, as in Quine's, isn't nearly as fundamental to the classical
meta-logical results as standard presentations make it seem like it must
be. Quine's presentation of the incompleteness theorem in "protosyntax"
seems to me to make it much clearer what's actually going on with
respect to incompleteness than do the usual proofs, which involve too
many different ideas. (Grzegorczyk's proof fills in the details and does
a lot else besides.) But the question asked was about the *strength* of
theories adequate for syntax, so the difference didn't seem important.

Richard Heck

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Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University