[FOM] axiomatizations of PA

[Dennis E. Hamilton]

I cannot address the historical situation.   However, your student might
find interesting discussions on the Mathematics Stack Exchange.  In
rummaging around the Internet I found this possibly-relevant topic:
<https://math.stackexchange.com/questions/2256637/why-are-we-using-first-ord
er-logic-and-how-to-fix-pa>.

The one answer is almost more informative than the question.

I'm also puzzled by what is meant by "variety of axiomatizations of PA" when
PA is a specific axiomatization of what are regarded as the natural numbers.
There seems to be great agreement on what PA is, however it has converged
over time.  Perhaps the puzzlement is over alternative axiomatizations of
the natural numbers and the motivations for those?

It is fashionable to fancy set-theoretic representations such as finite von
Neumann ordinals.  I would have thought this to be a separate matter.  Is
this a question for your student?

Regards,

 - Dennis E. Hamilton


-----Original Message-----
From: fom-bounces at cs.nyu.edu <fom-bounces at cs.nyu.edu> On Behalf Of UCKELMAN,
SARA L.
Sent: Friday, December 6, 2019 06:11
To: Foundations of Mathematics <fom at cs.nyu.edu>;
women-in-logic at lists.rwth-aachen.de
Subject: [FOM] axiomatisations of PA

Today [I was asked] about the variety of axiomatisations of PA, some of
which use the notion of "natural number" directly in the axioms, while
others (e.g., the one used by Goldstern & Judah) have axioms governing
each of the mathematical functions + induction.

Has anyone ever written on the development of axiomatisations of PA,
from a moderately historical, rather than mathematical, perspective?
[ ... ]