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 and , rather than absoluteness between and ? 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”.
Some of the first properties we learn about forcing notions are the notions of being atomless and being separative. Usually any kind of analysis of these terms are left out, as “all forcings we care about are atomless and separative”, so this post will be dedicated to taking a slightly closer look at these properties.
One of the first forcing facts that we learn is that -closed forcings preserve all sequences of length . This is usually shown via distributivity, by showing that every -closed forcing is also -distributive, and that -distributivity is equivalent to the forcing not adding any new sequences of length . I will recall these facts here, and show how they relate to both and . Here is the axiom of choices, stating that we have choice functions for all sets injecting into , and is the axiom of dependent choices, saying that every pruned tree of height at most has a branch.
The previous two posts was dedicated to stating, explaining and applying a certain result in core model theory, the PD dichotomy, without using any inner model theory at all. This post is then the final post in this short series in which we’ll actually prove the dichotomy. This blog series, and especially the following proof, grew out of some work with Stefan Mesken.
This is a continuation of my last post, in which I argue that core model theory can provide tools which other set theorists can use without having indepth knowledge of their proofs. The tool I chose was the following core model dichotomy, and in this post we’ll dig into a couple of examples in which we apply the dichotomy to various areas of set theory.
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.
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.