The Universe

[matthias]

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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