[FOM] Ruling-out Nonstandard Models of 1st-Order PA

[Roger Bishop Jones]

On Monday 23 October 2006 21:00, rtragesser at mac.com wrote:

> A natural question is: Formal First-Order Peano Arithmetic
> [abbrv. PA henceforth]  has non-standard models; what informal
> ideas are sufficiently clear and distinct, such that (formal)
> PA supplemented by these informal ideas with "informal rigor"
> rules out the nonstandards models ?

I was provoked into considering a related question a few years 
ago when I read a paper by Harty Field entitled:

"Which Undecidable Mathematical Sentences Have Determinate Truth 
Values?"

In which Field argues ultimately that appeal to certain 
metaphysical principles ("cosmological hypotheses") is both 
sufficient and necessary to make the truth values of arithmetic 
sentences determinate.

The idea that the sentences of arithmetic were less certain than 
the metaphysical conjectures, or could be clarified by appeal to 
them seemed to me highly implausible so I spent a little time on 
a refutation.

I set about showing a conjecture which has as a corollory (what 
is probably a well known result) that the only w-consistent 
complete extension of PA is "true arithmetic".  My conjecture 
was false but Rob Arthan came up with a proof of the corollory.

My notes on Fields paper are at:

http://www.rbjones.com/rbjpub/philos/bibliog/field97.htm

and Rob Arthan's proof is at:

http://www.rbjones.com/rbjpub/logic/rda40/rda40.htm

Of course this does not settle your question, for even "true 
arithmetic" has non-standard models, but it is nice to know that 
this leaves no questions expressible in first order logic 
indeterminate in truth value.

My own present opinion concerning the models, is that the 
standard model of arithmetic is so simple and intuitive that 
there is very little hope of finding any better place to start 
from.

Roger Jones