[FOM] Indefinitely large finite sets

Stephen G. Simpson sgslogic at gmail.com
Thu Dec 6 10:55:21 EST 2018

On Wed, Dec 5, 2018 at 6:47 PM Matthias Eberl <matthias.eberl at mail.de>

> I like to very shortly explain the finitistic approach of Lavine and
> Mycielski, since I think that it is not well known and I also think that it
> is interesting.
>  [...]
 (I don’t like the word “potential infinite”, since it suggests that it is
> a kind of infinity).

Thank you for explaining and calling attention to the ideas of Mycielski
and Lavine.  I am interested in but not very familiar with these ideas.

Some possibly relevant publications of mine are: "Partial realizations of
Hilbert's Program," "Toward objectivity in mathematics," "An objective
justification for actual infinity?," and recently "Foundations of
mathematics: an optimistic message."  These papers touch on certain aspects
of reverse mathematics which provide grounds for optimism.

As regards terminology, I prefer "potential infinity" to "indefinitely
large finite set," because it seems to me that the latter phrase is
confusing and misleading.  I view potential infinity as a species of
infinity, not as a species of finiteness.  In my view, a potential infinity
-- e.g., the revolutions of the earth around the sun -- should not be
regarded as finite (i.e., as a completed finite totality), nor should it be
regarded as an actual infinity (i.e., as a completed infinite totality).
In order to understand the real world, we need the concept of potential
infinity.  These distinctions go back to Books M and N of Aristotle's

Now that I think of it, my example of the earth revolving around the sun
seems especially timely.  Happy New Year!

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.math.psu.edu/simpson
email: sgslogic at gmail.com
telephone: 814-404-6176
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