[FOM] How much of math is logic?

[Dennis E. Hamilton]

It pains me to see this statement:

    I still think that even the axioms of successor are not 
    really logically true, for they are false 
    in all finite models.

It seems far more appropriate (and, yes, logical) to observe that there are
no finite models of PA, and that is a logical consequence of the axioms of
successor.  I think we can be farely confident that Peano intended that
conseqence.

I am also confused by the presumption of set theory in much of this
discussion, when that is certainly not essential to FOL= along with the
axioms (schemes) of Peano Arithmetic and with no other assertions with
regard to the domain of discourse.

I suspect that PA is of little utility for significant chunks of mathematics
(i.e., real analysis and calculus), while it is already too much for those
logicians concerned about predicativity.  And if we were to assert that
broader theories could simply be coded in PA, we should certainly deserve
the shrieks and laughter of mathematicians as they ran from our sight.

With regard to the commonplace formulations that start with set theory and
encode/represent PA (e.g., using von Neumann ordinals), we now seem to have
wandered even farther away from safely-logical territory.

I have no response for the question at hand.  I am simply concerned that we
need to be more precise in identifying the territory that is presumed as
solely logical.

 - Dennis

-----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: Sunday, March 04, 2007 04:48
To: Foundations of Mathematics
Subject: Re: [FOM] How much of math is logic?

Let us take as our logic two-sorted FO logic, with separate variables for 
numbers and classes.

[ ... ]

This is as close to logicism I can get - though I still think that even 
the axioms of successor are not really logically true, for they are false 
in all finite models.

[ ... ]
   
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