# Characterisation of Woodins in mice

I’ve previously mentioned an interesting characterisation of Woodin cardinals, that they are more or less characterised by the fact that we can do genericity iterations with them, i.e., that we can “capture reals” when Woodin cardinals are present. This is an exercise in the core model induction book, and Stefan Mesken recently found a way to solve this (see his solution here), which I’ll be presenting here.

# Git repositories

I’ve recently switched to Git, which among other things makes it possible for me to share most of my set theory documents (like theses, projects, notes and diagrams). This of course includes the pdfs, but also the tex files. So if there’s anything you might use, feel free to grab it. Also, if you spot any mistakes and/or typos, you can fix them if you want and make a pull request. Exciting!

Here’s a link to all my public repositories, which includes the following:

• Diagrams, containing all the diagrams in my diagrams section on this website and more;
• BSc-thesis, containing my bachelor’s thesis on the constructible universe L;
• Notes-and-talks, containing a lot of notes I’ve written and talks I’ve given over the years;
• Determinacy-project, containing a project I did during my master’s on determinacy;
• MSc-thesis, containing my master’s thesis on inner model theory;
• Basic-tex-template, which I use whenever I want to write a new paper or note.

Hope you find it useful.

# 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.

# 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”.

# Separating atoms

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.

# 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.