The Universe

[Joe Shipman]

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

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