The Universe (matthias)

[Øystein Linnebo]

Tim Chow asks the good question of whether there is a statement which can
be proved by someone who regards the natural numbers as an actual infinity
but not by someone who regards them merely as a potential infinity.

On the analysis that Shapiro and I offer, the answer has two parts.

First, while the actualist is entitled to full second-order arithmetic, the
potentialist is not, at least not on a combinatorial conception of the
classes of numbers (in which case every such class that is ever generated
is finite).

Second, consider a potentialist who is *strict*, roughly in the sense that
she insists not only that *every object* is generated at some stage or
other, but also that *every truth* is accounted for at some stage or other.
On the analysis that Shapiro and I provide, a strict potentialist is only
entitled to intuitionistic logic, not classical -- with obvious
consequences already for first-order arithmetic.

Oystein




> Date: Sun, 12 Jul 2020 12:47:03 -0400 (EDT)
> From: "Timothy Y. Chow" <tchow at math.princeton.edu>
> To: fom at cs.nyu.edu
> Subject: Re: The Universe (matthias)
> Message-ID: <alpine.LRH.2.21.2007121241360.28666 at math.princeton.edu>
> Content-Type: text/plain; charset=US-ASCII; format=flowed
>
> Oystein Linnebo wrote:
> >
> > In a recent paper, "Actual and potential infinity" (Nous 2018),
> > https://onlinelibrary.wiley.com/doi/abs/10.1111/nous.12208, Stewart
> > Shapiro and I take issue with the skeptical attitude illustrated by
> > Niebergall (as Adrian Mathias reminds us). Using the resources of modal
> > logic, we show how to articulate a clear and interesting distinction
> > between actual and potential infinity, which can be applied both to
> > arithmetical and set-theoretic potentialism.
>
> I found a draft version here---
>
> https://www.duo.uio.no/handle/10852/59120
>
> ---and skimmed it.  Here's a question.  Suppose Alice believes in actual
> infinity but Paul believes only in potential infinity.  Is there an
> example of a mathematical statement that both Alice and Paul regard as
> meaningful but that only Alice can prove, because the proof relies in an
> essential way on the assumption that an actual infinity exists?
>
> Tim
>
>
>
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