The Universe (matthias)

[Arnon Avron]

A question: Why should someone who is actualist about countable infinity be entitled to full second-order arithmetic?

Arnon
________________________________
From: FOM <fom-bounces at cs.nyu.edu> on behalf of Øystein Linnebo <linnebo at gmail.com>
Sent: Tuesday, July 14, 2020 5:38 PM
To: fom at cs.nyu.edu <fom at cs.nyu.edu>
Subject: Re: The Universe (matthias)

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<mailto:tchow at math.princeton.edu>>
To: fom at cs.nyu.edu<mailto:fom at cs.nyu.edu>
Subject: Re: The Universe (matthias)
Message-ID: <alpine.LRH.2.21.2007121241360.28666 at math.princeton.edu<mailto:alpine.LRH.2.21.2007121241360.28666 at math.princeton.edu>>
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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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