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

- On set-theoretic mereology as a foundation of mathematics, Oxford Phil Math seminar, October 2018 September 21, 2018 Joel David Hamkins
- Parallels in universality between the universal algorithm and the universal finite set, Oxford Math Logic Seminar, October 2018 September 21, 2018 Joel David Hamkins
- The rearrangement number: how many rearrangements of a series suffice to validate absolute convergence? Warwick Mathematics Colloquium, October 2018 September 20, 2018 Joel David Hamkins
- The Stable Core September 11, 2018 Victoria Gitman
- The propagation of error in classical geometry constructions September 10, 2018 Joel David Hamkins

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