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

- Lectures on the philosophy of mathematics, Oxford, Michaelmas term 2019 September 20, 2019 Joel David Hamkins
- Methods in Higher Forcing Axioms: The inevitable conclusion September 17, 2019 Asaf Karagila
- Weeknote 2019/37 September 14, 2019 Peter Krautzberger
- Carnival of Math No. 173 September 12, 2019 Peter Krautzberger
- Being a refugee September 11, 2019 Dave Sixsmith – I am a mathematician, not a calculator

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