My first paper was “An incompleteness theorem for $\beta_n$ models” with Stephen Simpson [1]. It’s a short paper, but the idea is very pretty. We know that the incompleteness theorem implies there are strange models of arithmetic, but these models often seem mysterious, and it’s hard to see what useful properties they can have. But now suppose that a theory of the form $A+B$ meets the hypotheses of the incompleteness theorem, and moreover this theory proves its own consistency, so that $A+B$ is inconsistent. It follows that if $A$ is true (that is, true in the standard model) then $B$ must be false. In this way, we can use the incompleteness theorem to prove facts about the standard model rather than about nonstandard ones. The idea is originally due to Harvey Friedman in his thesis, I believe.

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### Boole’s Rings

- Modern class forcing November 13, 2019 Victoria Gitman
- Knaster and friends II: The C-sequence number October 23, 2019 Assaf Rinot
- (with Y. Hayut) Perfect Subtree Property for Weakly Compact Cardinals October 11, 2019 Sandra Müller
- On conjugacy problems for graphs and trees October 11, 2019 Samuel Coskey
- Ground model definability in ${\rm ZF}$ October 10, 2019 Victoria Gitman

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