[FOM] First-order arithmetical truth

[Arnon Avron]

On Tue, Oct 17, 2006 at 11:59:40PM +0100, V.Sazonov at csc.liv.ac.uk wrote:
 
> > If you lack the ability to distinguish the intended model of PA from
> > another model,
> 
> Excuse me Timothy, please. Do YOU have this ability?
> 
> Arnon, may be you have?
> 
> Does anybody here have this ability?

I certainly do. So does everybody else here (well, almost everybody).
What is more: so does almost every person from some age onwards.
Thus everybody who uses the notions of "ancestor" and "descendant"
distinguishes the intended model of PA from all the others. Take
my word: when God promises Avraham in the old testiomony
to give this or that to Avraham's descendants, he meant only 
his standard ones, and no reader has ever expected him to keep 
the promise to any nonstandard descendant (especially that God 
seems to find difficulties in keeping his promises even to 
the standard descendants...).

Do you pretend not to understand the notion of a descendant, and
can you imagine yourself having nonstandard descendants?
(what can be the possible meaning of this??)

  But the whole question is wrong. You emphasize in your question
"models of PA". But what is so special about PA? The only
interest in PA is due to the fact that it codifiersd a part of what
we know to be true about the natural numbers ("the intended model").
Conceptually and historically, the structure of the natural 
numbers precedes PA, and nobody would have dreamt to introduce
PA otherwise (I wonder whether even you really take PA+CON(PA)
and PA+not-CON(PA) to be of the same significance and interest,
even though I dont see what possible reason you might have to prefer 
one over the other). Now PA, being a first-order theory, has more
than one model: the intended one, and many others. So what?
So do many other theories, like Q (which is weaker) and PA+CON(PA)
(which is stronger). Why should nonstandard models of PA
be anymore interesting or better candidates to be "alternatives"
to the real natural numbers than nonstandard models of Q
or of PA+CON(PA)? Well, perhaps you answer that it is the
most natural first-order approximation of Dedkind-Peano
second-order  categorical axiomatization. Even so, attaching
great significance to nonstandard models of PA just means 
attaching unjustified philosophical importance to first-order
axiomatizations, despite the fact that first-order languages
are simply too weak for characterizing interesting structures. 

  For many years I maintain that the appropriate language
for formalizing logic and mathematics is neither the first-order 
language nor the second-order one. The first is too weak for 
expressing what we all understand, the second involves too strong 
ontological commitments. The adequate language
is something  in the middle: what is called "ancestral logic"
in Shapiro's book "Foundations without Foundationalism". This logic 
is equivalent to weak second-order logic (as is shown
in Shapiro's book, as well as in  his chapter in Vol. 1 of the 
2nd ed. of the Handbook of Philosophical logic). However, I prefer 
the "ancestral logic" version, because the notion of "ancestor" 
is part of everybody's logic, 100% understood also by 
non-mathematicians. Needless to say, the Natural numbers 
can be categorically characterized in this language. Of course,
unlike first-order logic, there is no r.e. axiomatization
of "ancestral logic". This is indeed unfortunate, but nobody promises
us that everything should be according to our wishes...

Arnon Avron