The Universe (matthias)

[Øystein Linnebo]

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.

Best regards,
Øystein


> Date: Wed, 08 Jul 2020 06:39:25 +0000
> From: matthias <matthias.eberl at mail.de>
> To: fom at cs.nyu.edu
> Subject: Re: The Universe
> Message-ID: <em112cd31c-6cf6-4411-91c2-426e6b15b423 at laptop-482omb9d>
> Content-Type: text/plain; charset="utf-8"; Format="flowed"
>
> This debate is in its essence very similar to that between potential and
> actual infinite, basically between a dynamic and a static concept.
> Hamkins calls the multiverse view potentialism e.g.
>
> https://www.winterschool.eu/files/1040-Set-theoretic_potentialism_12048749381.pdf
> .
>
> Niebergall (https://philpapers.org/rec/NIEAOI) analyzed the notions of
> infinity, actual und potential, and finitism. One outcome is that there
> is no satisfactory way to formulate (purely syntactically) whether a
> theory uses a potential infinite and not an actual infinite. I think
> this transfers to the ZFC universe and multiverse debate as well. The
> adequate way to formulate this difference is in my opinion that of the
> interpretation of the universal quantifier. A potentialist view can only
> refer to some stage of the multiverse, not simply to "all". So the
> interpretation must have a kind of reflection principle in the sense
> that the universal quantifier refers to a stage that represents the
> whole multiverse.
>
> Kind regards,
> Matthias
>
> ------ Originalnachricht ------
> Von: "Joe Shipman" <joeshipman at aol.com>
> An: "Foundations of Mathematics" <fom at cs.nyu.edu>
> Gesendet: 08.07.2020 06:11:22
> Betreff: The Universe
>
> >I have been thinking about the debate between the ?Multiverse? and ?One
> Universe? viewpoints in Set Theory.
> >
> >It?s a bit hard to figure out exactly what their disagreement is, well
> enough to state what would count as ?evidence? for one view or the other.
> >
> >I?m assuming both camps agree on ZFC, and only count as ?Universes? set
> or class models of ZFC which are well-founded, standard, and transitive, to
> sharpen the issues I care about.
> >
> >I?d also like to ignore distinctions between different models with the
> same theories. So we have exactly continuum-many theories consistently
> extending ZFC. Let S be this set of theories.
> >
> >Do ?One Universe? theorists and ?Multiverse? theorists have well-defined
> and opposing views on any statement about which elements of S have standard
> transitive set models or class models?
> >
> >Do ?One Universe? theorists believe that the element of S that is the
> theory of ?the? universe has a set model?
> >
> >Is there any element of S which ?One Universe? theorists are sure is not
> the theory of ?the? Universe, but which has a standard transitive set or
> class model?
> >
> >What is an example of a statement which, if proved in ZFC, might persuade
> some members of one camp that the other camp was correct?
> >
> >? JS
> >
> >Sent from my iPhone
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