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.

Continue reading

### Boole’s Rings

- Faculty respondent to paper of Ethan Jerzak on Paradoxical Desires, Oxford Graduate Philosophy Conference, November 2018 November 9, 2018 Joel David Hamkins
- (with J. Aguilera) The consistency strength of long projective determinacy November 6, 2018 Sandra Müller
- The Axiom of Determinacy implies Dependent Choices in mice November 5, 2018 Sandra Müller
- 6 Thoughts on accessibility of equation layout October 28, 2018 Peter Krautzberger
- Cohen's Oddity October 20, 2018 Asaf Karagila

### Comments on Boole’s Rings

- Comment on Lectures on the Philosophy of Mathematics, Oxford, Michaelmas 2018 by Joel David Hamkins November 9, 2018 Comments for Joel David Hamkins
- Comment on Lectures on the Philosophy of Mathematics, Oxford, Michaelmas 2018 by Artem Kaznatcheev November 8, 2018 Comments for Joel David Hamkins
- Comment on Alan Turing, On computable numbers by Joel David Hamkins October 13, 2018 Comments for Joel David Hamkins
- Comment on Alan Turing, On computable numbers by Lorenzo Cocco October 13, 2018 Comments for Joel David Hamkins
- Comment on Alan Turing, On computable numbers by Joel David Hamkins October 13, 2018 Comments for Joel David Hamkins