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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Applied core model theory I

Inner model theory and core model theory might seem like their own niche in set theory, where you have to invest hundreds of hours just to get a glimpse of what’s going on. But behind all the complicated theory there are theorems in inner model theory which can be applied in many contexts with minimal background knowledge of the intricate technicalities appearing in their proofs. In this and the next couple of blog posts I’ll introduce one such theorem, explain how to use it, do a few mainstream set theory applications of it, and also provide a proof of it. Everything aside from the proof should hopefully be accessible to set theorists who aren’t inner model theorists.

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Antichains and closure properties

There are many different properties that forcings can have, whose consequences are usually well-known. As an example, intuitively, closure properties of forcings yield preservation of cardinals below, and antichain properties yield preservation of cardinals above. But these properties seem mostly to be studied individually, so Stamatis Dimopoulos and I set out to find these folklore results about which combinations of closure properties and antichain properties can consistently hold.

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A categorical approach to forcing?

When doing set theory we tend to take pride in the fact that it’s something unique, something that is doing things in a very different way from the rest of mathematics. Take things like transitive sets, forcing extensions, elementary embeddings as well as syntactic considerations in results like Shoenfield absoluteness. But there are times when we might be better off by borrowing terminology, and maybe even results, from other fields. In this post I will just give a potential example of such a thing: namely, considering the structural side of forcing from a categorical point of view.

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A universe of hierarchies

When doing set theory (or mathematics in general) we’re working inside some universe, usually denoted by V. Since we can’t work with everything there is (in a first-order way), we resort to working with initial segments of V. The confusion then arises, since what do we mean by an initial segment? Some prefer to work with the “rank-hierarchy” V_\alpha and others prefer to work with the “hereditary cardinality hierarchy” H_\kappa. It gets even worse if we’re working in Gödel’s constructible universe L, since then we also got Gödel’s L_\alpha hierarchy and Jensen’s J_\alpha hierarchy. How do we picture these hierarchies? What are their relation to each other?

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