Mathematics with the potential infinite

[Stephen G. Simpson]

Philosophically, the distinction between potential infinity and actual
infinity is of great importance for mathematics.  So far as I know, the
first substantial discussion of this distinction is in Books M and N of
Aristotle's Metaphysics.

In modern foundational studies, Hilbert's Program plays a key role.
Hilbert's Program is, to give a "finitistic" proof of the consistency of
"infinitistic" mathematics.   One can argue about how to formalize what we
mean by "finitism," but philosophically I believe that "finitism" is
essentially a commitment to eschew actual infinities and reason solely with
potential infinities.

William W. Tait, in his 1967 paper "Constructive reasoning" and his 1981
paper "Finitism," argued that the "finitism" in Hilbert's Program is
correctly formalized as PRA (= Primitive Recursive Arithmetic).  Accepting
this argument of Tait, I argued in my 1988 paper "Partial realizations of
Hilbert's Program" that Hilbert's Program has been largely successful,
despite Gödel.  My argument is based on the study of certain formal systems
which, while not finitistic, are "finitistically reducible" in the sense
that they are conservative over PRA for finitistically meaningful
sentences.  The point is that these systems are mathematically powerful in
that they account for much of classical analysis, algebra, geometry, etc.
But of course "PA is consistent" is not provable in PRA, and this seems to
be relevant for Timothy Y. Chow's question.

A popular exposition of my ideas on this is my 2020 paper "Foundations of
mathematics: an optimistic message."  The paper consists of the text and
slides of a talk for a *general audience*, i.e., an audience of
non-mathematicians.

Stephen G. Simpson
Research Professor

Department of Mathematics
1326 Stevenson Center
Vanderbilt University
Nashville, TN 37240, USA

office: 804 Central Library
Vanderbilt University

web: www.personal.psu.edu/t20
email: sgslogic at gmail.com
telephone: 814-404-6176



On Thu, Jan 26, 2023 at 10:56 PM Timothy Y. Chow <tchow at math.princeton.edu>
wrote:

> Matthias wrote:
>
> > for several years I have been engaged in developing mathematics solely
> > with the concept of the potential infinity. Two publications of mine are
> > now available.
>
> This is interesting.  The topic of potential infinity has come up on FOM
> before a few times, e.g., in March 2015---
>
> https://cs.nyu.edu/pipermail/fom/2015-March/
>
> ---and in July 2020, when you made a few comments:
>
> https://cs.nyu.edu/pipermail/fom/2020-July/
>
> One question I have in general for any account of potential infinity is
> this: Can you give examples of statements that are provable using "actual
> infinities" but that are not provable using "potential infinity" only?
> (Here, I exclude statements that don't even make sense unless you assume
> actual infinities.)  As a specific test case of interest, can you prove
> "PA is consistent" using only only potential infinity?
>
> Tim
>
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