This was an online talk 15 April 12:15 for the CUNY Set Theory Seminar. Held on Zoom at 876 9680 2366.

**Abstract.** I shall give a general introduction to urelement set theory and the role of the second-order reflection principle in second-order urelement set theory GBCU and KMU. With the *abundant atom axiom*, asserting that the class of urelements greatly exceeds the class of pure sets, the second-order reflection principle implies the existence of a supercompact cardinal in an interpreted model of ZFC. The proof uses a reflection characterization of supercompactness: a cardinal $\kappa$ is supercompact if and only if for every second-order sentence $\psi$ true in some structure $M$ (of any size) in a language of size less than $\kappa$ is also true in a first-order elementary substructure $m\prec M$ of size less than $\kappa$. This is joint work with Bokai Yao.

See the article at: Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal

Unfortunately I can’t make this, but it looks super interesting. Question:

1. Can I see slides/paper?

2. Have you considered how this will relate to Chris Menzel’s theories of `wide sets’?

Paper is currently still in progress (frantically still writing!), but should be available before too long. I don’t know much about Menzel’s theory, but I’ll look into it. Sounds superficially like it could be related, since a fundamental feature of the abundant atom axiom is that the universe is very wide in comparison with its height.

Update:paper is now available at: http://jdh.hamkins.org/second-order-reflection-with-abundant-urelements/Seconding Neil’s second question (hehe), there seems to be some interest in the modal set theory and metaphysics circles in the following principle: for each cardinal $\kappa$, it is possible for there to be $\kappa$ many urelements. I learned about this from my undergrad professor Gabriel Uzquiano in his article titled Recombination and Paradox, where he attributes it to Sider and Menzel. Looking forward to the talk!

Hi Jason! I’m so glad you’re also aware of this potential connection between our work and modal metaphysics! Two quick comments:

(1) The abundant atom axiom will go far beyond the principle you mentioned (plz come to the talk for its formal definition 🙂 )

(2) The recombination paradoxes in Uzquiano’s paper appeal to the limitation of size principle (LS). But LS is a very special second-order choice principle in KMU: it doesn’t follow from the second-order choice principle (RP_2) assuming large cardinals. And what is really surprising is the exact strength of KMU + RP_2 + ~LS….

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