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

- Oxford University, Professor of Logic & Sir Peter Strawson Fellow, University College Oxford May 17, 2018 Joel David Hamkins
- My teaching advice and resources (so far) May 12, 2018 Mike Pawliuk – Mathematics
- Critical Cardinals May 10, 2018 Asaf Karagila
- Knaster and friends I: Closed colorings and precalibers April 26, 2018 Assaf Rinot
- Set-theoretic potentialism and the universal finite set, Scandinavian Logic Symposium, June 2018 April 22, 2018 Joel David Hamkins

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