[FOM] axiomatizations of PA

[Dennis E. Hamilton]

It is gratifying to see that Cohen's book is now available in a Dover reprint 
of the typescript (Kindle edition not recommended by reviewers though).  I 
look forward to arrival of my inexpensive print copy.

Thanks to the observation by Lukasz T. Stepien, I was able to track down 
Gabriele Lolli and see that his Chapter on "Peano and the Foundations of 
Arithmetic" is available on ResearchGate.  It is a delightful account that 
demonstrates how much foundational questions were in the air at the time.  I 
can't help but mention two nuggets that caught my eye.

1. That having an infinite model leads to the existence of many such models is 
anticipated, as is the "problem" of the possibility of members of N that are 
not part of the inductive progression.

 2. There is polite mention of Russell's rather indignant irritation over the 
lack of a metaphysical commitment to the fixed unit (1 or 0) of ordinary 
arithmetic.  This wanting of a strict interpretation of our sense of something 
in nature (even a Platonic reality) is not unlike the distress some folks have 
over the empty set and the denumerable infinite (and a common misunderstanding 
of material implication as well).

There is much more on the interplay with Dedekind, Frege, and predecessors who 
set groundwork for PA.

All of the matters and concerns have been refined again and again in the 
ensuing century, yet in retrospect the roots of the convergence to today's 
framing (and enduring concerns) are evident.

 - Dennis

-----Original Message-----
From: fom-bounces at cs.nyu.edu <fom-bounces at cs.nyu.edu> On Behalf Of Joe Shipman
Sent: Saturday, December 7, 2019 13:01
To: Foundations of Mathematics <fom at cs.nyu.edu>
Subject: Spam (12.623):Re: [FOM] axiomatizations of PA

?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'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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