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

- What a long strange trip it’s been… April 25, 2017 Asaf Karagila
- The MacLane class and the Eremenko–Lyubich class April 21, 2017 Dave Sixsmith
- The inclusion relations of the countable models of set theory are all isomorphic April 17, 2017 Joel David Hamkins
- All countable models of set theory have the same inclusion relation up to isomorphism, CUNY Logic Workshop, April 2017 April 13, 2017 Joel David Hamkins
- The definable cut of a model of set theory can be changed by small forcing April 10, 2017 Joel David Hamkins

### Comments on Boole’s Rings

- What a long strange trip it’s been… April 25, 2017 Asaf Karagila
- Comment on The inclusion relations of the countable models of set theory are all isomorphic by All countable models of set theory have the same inclusion relation up to isomorphism, CUNY Logic Workshop, April 2017 | Joel David Hamkins April 17, 2017 Comments for Joel David Hamkins
- Comment on Set-theoretic mereology by The inclusion relations of the countable models of set theory are all isomorphic | Joel David Hamkins April 17, 2017 Comments for Joel David Hamkins
- Comment on Set-theoretic mereology by The countable models of set theory all have isomorphic inclusion relations, CUNY Logic Workshop, April 2017 | Joel David Hamkins April 13, 2017 Comments for Joel David Hamkins
- Comment on The definable cut of a model of set theory can be changed by small forcing by Joel David Hamkins April 11, 2017 Comments for Joel David Hamkins