Kameryn's Web Log

This post arose from thinking about some questions Joel David Hamkins raised on twitter, in concert with some thoughts that’d been kicking around my head for a while.

Bi-interpretability is commonly taken to say that two theories have equivalent content and working in one over the other is a matter of personal preference with no mathematical difference. I think this may be too hasty a conclusion, and this blog post is an attempt to explicate a possible counterexample, which I claim no essential novelty in raising.

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This blogpost arose from me looking at Kanovei and Shelah’s paper “A definable nonstandard model of the reals”, and thinking about ultraproducts over spaces more general than powerset boolean algebras.

Everyone is familiar with ultrapowers, but

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There’s a piece of pedagogical practice in maths which I hate.

Sets are ubiquitously useful in mathematics, so any student of mathematics will have to learn about them. The first definition given is usually something like the following, let’s call it the proto-definition:

A set is a well-defined collection of objects.

So, for example, there is the set of all real numbers, the set of all polynomials with at least one root, the set of all sets of real numbers, and so on. If you have some property $P(x)$ which can be true or false of objects $x$, then there is a set $\{x : P(x)\}$ of all objects $x$ for which $P(x)$ is true.

This picture of sets is good for a first approximation, but it has a serious flaw.

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Today many academics are participating in #ShutDownSTEM, forgoing our usual academic work to instead work on how to improve the current dismal state of STEM (Science, Technology, Engineering, and Mathematics) with regard to racism, especially as it harms black people. This is part of a larger reaction to the uprising sparked by the police murder of George Floyd. If, as we say in the slogan, black lives matter, then black lives matter in the academy and we should back up our words with action.

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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.

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(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.

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Consider the following set-up, 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.

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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.

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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 end-extension to a model of $\mathsf{ZFC} + V = L$.

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