Descriptive inner model theory – an overview

Inner model theory is all about constructing canonical inner models of set theory that inherits the complexity — i.e. the large cardinals — of the universe. In “classical”, or “pure”, inner model theory it’s clear that there’s been a lot of partial progress towards this goal, as the programme has resulted in explicit constructions of inner models inheriting a lot of the large cardinals present in V. But with the emergence of descriptive inner model theory this is suddenly not as clear. Where are the inner models containing large cardinals? I’ll do my best to give an overview of how this is accomplished and also how large cardinal theories, determinacy theories and arbitrary theories of interest (like forcing axioms) interact with each other.

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Three paths to weak compactness

Weakly compact cardinals have a great variety of properties, which are all interesting enough to warrant independent study. These properties include threadabilityreflection and Mahlo properties. Studying these features in isolation leads to interesting (non-)interactions and gives us three distinct hierarchies of large cardinal notions below weakly compacts in terms of direct implication, where in terms of consistency strength two of the hierarchies simply collapse.

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Consistency strength of forcing axioms

Previously I’ve only been talking about large cardinals and determinacy theories as if they were the only consistency hierarchies around. There is another important class of axioms, which has the added benefit of being, dare I say it, more useful to mathematicians not working in set theory. The reason for this is probably that these forcing axioms have a handful of consequences of a non-set theoretic nature, making them easier to apply in (mathematical) practice. When it comes to the consistency strength of these axioms though, things get a lot more hazy: we know very little about the strength of (almost all of) these axioms. I’ll introduce these axioms here and state what is known to date.

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Limitations of ZFC determinacy

I was recently playing a (set-theoretic) game and the question of whether it was determined slowly emerged. As I was working in a ZFC context, most of the determinacy results were of no use to me, so I tried to investigate how much we really know about ZFC determinacy. Of course we can’t have full determinacy (AD), but how about definable variants, where we alter both the objects played and the length of the game?

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HODs of models of determinacy

HOD is the proper class of all sets x such that both x and all the elements of the transitive closure of x are definable using ordinal parameters. HOD is a model of ZFC, but in general its structure is not really known. In the late 90’s it was shown by Steel and Woodin that \textsf{HOD}^{L(\mathbb R)} exhibits mouse-like behaviour, and since then there’s been a great interest in finding the HODs of other models than L(\mathbb R). I’ll here give a (non-exhaustive) overview of both which HODs have been shown to have this mouse-like structure and also explain the general strategy used so far in finding these mice.

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