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

- Towards a general method for accessible content trees or: deep aria-labels for equations revisited January 13, 2019 Peter Krautzberger
- Forcing as a computational process, Cambridge, Februrary 2019 January 5, 2019 Joel David Hamkins
- Foundations of analysis January 1, 2019 Samuel Coskey
- Matrices for classical groups January 1, 2019 Nick Gill
- Logic and set theory January 1, 2019 Samuel Coskey

### Comments on Boole’s Rings

- Comment on The subseries number by The rearrangement and subseries numbers: how much convergence suffices for absolute convergence? Mathematics Colloquium, University of Münster, January 2019 | Joel David Hamkins January 10, 2019 Comments for Joel David Hamkins
- Comment on Souslin trees at successors of regular cardinals by saf December 19, 2018 Comments for Assaf Rinot
- Comment on A new proof of the Barwise extension theorem, without infinitary logic, CUNY Logic Workshop, December 2018 by Joel David Hamkins December 16, 2018 Comments for Joel David Hamkins
- Comment on A new proof of the Barwise extension theorem, without infinitary logic, CUNY Logic Workshop, December 2018 by Joel David Hamkins December 15, 2018 Comments for Joel David Hamkins
- Comment on A new proof of the Barwise extension theorem, without infinitary logic, CUNY Logic Workshop, December 2018 by Athar December 15, 2018 Comments for Joel David Hamkins