The Universe (matthias)

[Øystein Linnebo]

Let me begin with a more precise statement of the claim that prompted
Arnon's question. Consider a theorist who accepts a combinatorial
conception of classes, i.e. a class as a completely arbitrary collection of
previously available objects. Claim: If this theorist is also an actualist
about the natural numbers, then she'll be entitled to full second order
arithmetic, whereas she will not be so entitled if she is merely a
potentialist about the naturals (for the reason I gave in my previous
message). Now, Arnon's question correctly reminds us that not everyone will
accept the antecedent, i.e. accept a combinatorial conception of classes.
Predicativism provides a famous and important example (as Arnon's own work
has helped make clear).

Incidentally, predicativism raises the possibility of its own kind of
potentialism, not (in the first instance) about the natural numbers, but
about *sets of natural numbers*. Indeed, in his famous "Systems of
predicative analysis, I" (1964), Feferman writes: "we can never speak
sensibly (in the predicative conception) of the "totality" of all sets as a
"completed totality" but only as a *potential totality *whose full content
is never fully grasped but only *realized in stages*" (p. 2; his italics).

Oystein



On Wed, 15 Jul 2020 at 11:43, Arnon Avron <aa at tauex.tau.ac.il> wrote:

> 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>
> 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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