Talk: Level-by-level virtual large cardinals

I am giving an invited talk at the set theory seminar at the City University of New York, on Friday 15 February. I will be talking about virtual large cardinals, which grew out of my previous work on Ramsey-like cardinals. Here’s an abstract:

“A virtual large cardinal is (usually) the critical point of a generic elementary embedding from a rank-initial segment of the universe into a transitive M\subseteq V, as introduced by Gitman and Schindler (2018). A notable feature is that all virtual large cardinals are consistent with V=L, and they’ve proven useful in characterising several properties in descriptive set theory. We’ll work with the virtually \theta-measurable, \theta-strong and \theta-supercompact cardinals, where the \theta in particular indicates that the generic embeddings have H_\theta^V as domain, and investigate how these level-by-level virtual large cardinals relate both to each other and to the existence of winning strategies in certain games. This is work in progress and joint with Philipp Schlicht.”

 

Concrete and abstract

Abstraction is so common in mathematics that we usually don’t bat an eye when jumping between different levels of abstraction. There are many cases in which such an abstraction makes concepts clearer, as it cuts away all unneccesary bits of information, and also many cases in which something more concrete makes things easier to work with, as we have more information about how our objects of study actually look like. I’ll give a few well-known examples of this phenomenon from mathematics, and argue that it occurs in several (perhaps subtle) places in set theory as well.

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Shoenfield absoluteness and choice

Absoluteness of wellfoundedness and Shoenfield absoluteness are two absoluteness results in set theory that are both used incredibly often. But what if we want to apply the result to absoluteness between arbitrary models M and N, rather than absoluteness between V and L? It turns out that our models have to satisfy dependent choice in both absoluteness results, and in Shoenfield absoluteness we have to ensure that the models are of “similar height”.

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Closure, distributivity and choice

One of the first forcing facts that we learn is that \kappa-closed forcings preserve all sequences of length \kappa. This is usually shown via distributivity, by showing that every \kappa-closed forcing is also \kappa-distributive, and that \kappa-distributivity is equivalent to the forcing not adding any new sequences of length \kappa. I will recall these facts here, and show how they relate to both \textsf{AC}_\kappa and \textsf{DC}_\kappa. Here \textsf{AC}_\kappa is the axiom of \kappa choices, stating that we have choice functions for all sets injecting into \kappa, and \textsf{DC}_\kappa is the axiom of \kappa dependent choices, saying that every pruned tree of height at most \kappa has a branch.

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