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

- Kelley-Morse set theory does not prove the class Fodor Principle, CUNY Set Theory Seminar, March, 2019 March 20, 2019 Joel David Hamkins
- On splitting and splittable families March 7, 2019 Samuel Coskey
- In praise of Replacement March 6, 2019 Asaf Karagila
- Introduction to continuous logic and model theory for metric structures. March 1, 2019 Samuel Coskey
- Must there be numbers we cannot describe or define? Definability in mathematics and the Math Tea argument, Norwich, February 2019 February 19, 2019 Joel David Hamkins

### Comments on Boole’s Rings

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