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

- Talk: Canonical inner models and their HODs November 28, 2017 Sandra Uhlenbrock
- Quote by Henri Poincaré November 24, 2017 dcernst.github.io
- No representation without taxation. November 22, 2017 Joseph Van Name
- The universal finite set November 22, 2017 Joel David Hamkins
- A POW problem that can incentivize the energy efficient CNOT computer. November 18, 2017 Joseph Van Name

### Comments on Boole’s Rings

- Comment on No representation without taxation. by Joseph Van Name December 3, 2017 Comments for Joseph Van Name
- Comment on A strong form of König’s lemma by saf November 29, 2017 Comments for Assaf Rinot
- Comment on A strong form of König’s lemma by Ari B. November 29, 2017 Comments for Assaf Rinot
- Comment on No representation without taxation. by Jesse C. McKeown November 28, 2017 Comments for Joseph Van Name
- Comment on A forcing axiom deciding the generalized Souslin Hypothesis by saf November 27, 2017 Comments for Assaf Rinot