Kameryn's Web Log

The omegath hod may not satisfy choice
McAloon proved in the 70s that $\mathrm{HOD}^\omega$—i.e. the intersection of $\mathrm{HOD}$, the $\mathrm{HOD}$ of $\mathrm{HOD}$, and so on iterated out finitely far—may fail to satisfy choice. I decided to write up an exposition of his result because (1) it seems to be underappreciated and (2) I wanted to understand it and writing up an exposition of it was a good way to make that happen.

Recursive saturation, inner models, and forcing
(The content of this blog post arose from discussion with Victoria Gitman, in the context of work in a joint project with Michał Godziszewski and Toby Meadows.)
In the study of nonstandard models of set theory, one important property is that of recursive saturation. Originally due to Barwise and Schlipf, recursive saturation has proven to be a robust and powerful notion. Let me begin by reminding you of the definition.

Truth and potentialism
Consider the following setup, arising from discussion with Neil Barton over multiple coffees at Das Voglhaus. You think that classes are objects. And you think that if are committed to some object, then you should also be committed to the truth predicate about that object. So you should always be able to expand your classes by adding truth predicates. This can naturally be interpreted in a potentialist framework. What is the nature of the corresponding potentialist system? The answer, as I would like to explicate in this blog post, is that it depends.

Class Collection versus Class Choice
I’ve been thinking recently about different versions of Global Choice, which are equivalent in the presence of Powerset but not equivalent in its absence. As part of this I realized that this gives the answer to how different formulations of what I’ve variously seen called Class Choice/Class Collection/Class Bounding relate in the absence of Global Choice.

Every countable model of set theory endextends to a model of V = L
This theorem has popped up in my life a few times in the past week, and it’s one of the coolest results I know of, so I wanted to share it with the world.
Theorem (Barwise) Every countable transitive model of $\mathsf{ZF}$ has an endextension to a model of $\mathsf{ZFC} + V = L$.
(For blog posts from before summer 2018, see my old site.)