[FOM] Ruling-out Nonstandard Models of 1st-Order PA
Roger Bishop Jones
rbj01 at rbjones.com
Wed Oct 25 03:49:27 EDT 2006
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
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
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
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:
and Rob Arthan's proof is at:
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
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