[FOM] How much of math is logic?

[Dennis E. Hamilton]

I'm sorry, I'm still baffled.

I understand that the absence of finite models for PA is a metatheorem, if
you will.  Of course it is.  So is that a purely mathematical metatheorem,
or is it indeed "logical?"

I just don't see granting of existence to a finite model, which is certainly
external to what PA provides, as somehow more logical than that.  That
observation is hardly "in the system" either, it seems to me.

So how does one have a model in "all models" (of a theory, yes?) for which
an axiom does not hold?  Doesn't that mean there are sentences provable in
PA that are not satisfied in the finite [domain, yes?] structure and doesn't
that simply disqualify such a structure as a model of PA?  

Maybe this is just a clarification of language.  I am perfectly willing to
believe that I don't understand what a model is, or that "model" in this
conversation is not what a model is, say, on p.50 (Chapter A.2, the second
paragraph of section 2) in the Handbook of Mathematical Logic.  

I'm also puzzled, now that Panu mentions it, how "logically true" is meant
to help us with the question of how much of math is logic?  Does the math
that is logic have to be math that is logically true in this sense?  Is that
the crux of logicism?


 - Dennis

PS: I chose "utility" with some care, but not much authority [;<).  I
appreciate that it is not a logical claim and that "practically all" might
trump it.  That doesn't remove the finite-model question for me.  Nor,
actually, does it remove the question of having to commit to set theory.  Is
that commitment tacit in this discussion?  (I have no position, I just want
to know if it is tacit in this conversation, because I have difficulty
telling.)


-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
praatika at mappi.helsinki.fi
Sent: Monday, March 05, 2007 07:07
To: Foundations of Mathematics
Subject: Re: [FOM] How much of math is logic?

[ ... ]

It is perfectly standard to explicate "logically true" as "true in all
models", such that any formula that is false in some model, e.g., in some
finite model, is not logically true. In this sense, the axioms of successor
are, quite trivially, not logically true. 

Obviously it is a fact of (mathematical) logic (as a field of study), more
exactly, of model theory, that PA has no finite models. But certainly this
is not logically true; proving it requires some non-logical axioms. 

[ ... ]

> I suspect that PA is of little utility for significant chunks of
> mathematics (i.e., real analysis and calculus),

Wrong. For example ACA_0, which is a conservative extension of PA, is
sufficient for practically all ordinary analysis or calculus.