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

- Zoom into the classical Laver table fractals February 26, 2017 Joseph Van Name
- Open and clopen determinacy for proper class games, VCU MAMLS April 2017 February 25, 2017 Joel David Hamkins
- Computable quotient presentations of models of arithmetic and set theory, CUNY set theory seminar, March 2017 February 25, 2017 Joel David Hamkins
- Got jobs? February 19, 2017 Asaf Karagila
- 2017 Workshop in Set Theory, Oberwolfach February 19, 2017 Assaf Rinot

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