Mathematics with the potential infinite

[Øystein Linnebo]

Stewart Shapiro and I have been following the recent thread on this topic
with great interest and want to use this opportunity to summarize our take
on the topic, thereby hopefully also answering some of the questions that
have come up.

When analyzing the distinction between potential and actual infinity, it is
natural to invoke the resources of modal logic. Then the existence of a
potential and an actual (or "completed") infinity of natural numbers can be
expressed very naturally as follows, respectively:

(P) Necessarily for every number possible there is a larger one.
(A) For every natural number there is a larger one.

In [1], we show how this approach leads to a natural analysis of the
Aristotelian idea that there is a potential, but not an actual, infinity of
natural numbers.

While the modal language is useful for analyzing potential infinity, we
also define a translation from the non-modal language of ordinary
first-order arithmetic, based on translating "for all" and "exists" as
"necessarily for all" and "possibly exists", respectively. Given some
natural modal assumptions, we show in [1] that this translation yields a
faithful interpretation (or "mirroring") of first-order Peano Arithmetic in
our Aristotelian modal arithmetic. Thus, a potentialist who accepts this
modal arithmetic is justified in using first-order PA.

By contrast, we argue that an Aristotelian potentialist will not be
justified in using full second-order PA, if the second-order classes are
interpreted combinatorially (i.e., as arbitrary collections or pluralities
of numbers). Moreover, there is a stricter form of potentialism that is
only entitled to intuitionistic modal logic. Then the mentioned translation
yields a faithful interpretation of first-order Heyting Arithmetic.

Various extensions are possible too. In [2], we provide a modal analysis of
Feferman's claim [3, p.2; 4, p. 619] that a predicativist accepts a
completed totality of natural numbers but merely a potential totality of
sets thereof. Thus, predicativism too is naturally analyzed as form of
potentialism.

[1] Linnebo, Øystein & Shapiro, Stewart (2019). Actual and Potential
Infinity. Noûs 53 (1):160-191.
[2] Linnebo, Øystein & Shapiro, Stewart (2021). Predicativism as a Form of
Potentialism. Review of Symbolic Logic:1-32.
[3] Feferman, Solomon (1964). Systems of predicative analysis. Journal of
Symbolic Logic 29 (1):1-30.
[4] Feferman, Solomon (2005). Predicativity. In Stewart Shapiro (ed.),
Oxford Handbook of Philosophy of Mathematics and Logic. Oxford: Oxford
University Press. pp. 590-624.

Øystein (and Stewart)
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