[FOM] axiomatizations of PA

[Joe Shipman]

I think what is meant is a variety of axiomatizations of the same logical strength as PA. There is a good treatment of this in Paul Cohen’s book “Set Theory and the Continuum Hypothesis”.

— JS

Sent from my iPhone

> On Dec 6, 2019, at 10:13 PM, Dennis E. Hamilton <dennis.hamilton at acm.org> wrote:
> 
> 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?
> [ ... ]
> 
> 
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