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

- The 14th International Workshop on Set Theory in Luminy October 14, 2017 Assaf Rinot
- Virtual large cardinal principles October 12, 2017 Victoria Gitman
- Borel complexity theory and classification problems October 9, 2017 Samuel Coskey
- Dynamics in the Eremenko-Lyubich class October 8, 2017 Dave Sixsmith – I am a mathematician, not a calculator
- The problem with MathML as a web standard (part 4) September 29, 2017 Peter Krautzberger

### Comments on Boole’s Rings

- Comment on Ord is not definably weakly compact by Ali Enayat October 17, 2017 Comments for Joel David Hamkins
- Comment on Virtual large cardinal principles by Victoria Gitman October 16, 2017 Comments for Victoria Gitman
- Comment on Virtual large cardinal principles by Neil Barton October 16, 2017 Comments for Victoria Gitman
- Comment on Square principles by The 14th International Workshop on Set Theory in Luminy | Assaf Rinot October 15, 2017 Comments for Assaf Rinot
- Comment on Math for seven-year-olds: graph coloring, chromatic numbers, and Eulerian paths and circuits by Joel David Hamkins October 10, 2017 Comments for Joel David Hamkins