Categories of amenable embeddings and what canonicity in set theory cannot be
Monroe Eskew (University of Vienna)
In recent work with Sy Friedman, we explored a notion of width reflection: V ⊆ W are models of ZFC with the same ordinals, and there is an elementary embedding j : V → W such that j is an amenable predicate for W —equivalently, that j [x ] ∈ W for all x ∈ V . By Kunen’s Theorem, V ???? W , unless the critical sequence is cofinal in the ordinals. Although this notion looks like a very strong large cardinal property, it can be obtained from less than a measurable cardinal. The main focus of the paper is to explore what kinds of structures can be found among systems of such embeddings.
Given an ordinal δ, let Eδ be the category whose objects are all transitive models of ZFC of height δ and whose arrows are all elementary embeddings between these models. Let Aδ be the subcategory where we take only amenable embeddings as arrows. (It is easy to see that amenable embeddings are closed under composition.) Partial orders are naturally represented as categories where between any two objects there is at most one arrow. Let us say that a subcategory D of a category C is honest if whenever x and y are objects of D and thereisanarrowf :x→yinC,thenthereisoneinDaswell.
If there is a countable transitive model of ZFC of height δ satisfying that there is a proper class of large enough cardinals, then Aδ contains honest subcategories isomorphic to:
1. The real numbers.
2. An Aronszajn tree.
3. A universal countable pseudotree.
We would like to discuss the implications of the above result for the possible meanings of “canonical model” in set theory. It shows that whatever axioms we adopt that are consistent with having many large cardinals, the resulting theory cannot dictate a canonical way to build the universe along the ordinals. Epistemically, it seems to say that if we believe in a strong enough theory of sets, then our universe may be just one among a rich multiverse of models of the same height that are indistinguishable by first-order properties. Canonicity fails very badly, since we have literally a continuum of possibilities. We might have built the universe with more or fewer sets, with very little change in the total information. If we move through this multiverse by expanding or contracting our current universe, then this process is not canonical either, since we can always do that a little more or a little less. Furthermore, the tree constructions show that we may move into severely incompatible universes, which are nonetheless indistinguishable from the internal point of view.