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

- Bi-interpretation in set theory, Bristol, February 2020 January 22, 2020 Joel David Hamkins
- Fake Reflection January 20, 2020 Assaf Rinot
- Bi-interpretation in weak set theories January 16, 2020 Joel David Hamkins
- A Microscopic approach to Souslin-tree constructions. Part II January 7, 2020 Assaf Rinot
- Philosophy meets maths, Oxford, January 2020 January 7, 2020 Joel David Hamkins

### Comments on Boole’s Rings

- Comment on Bi-interpretation in weak set theories by Bi-interpretation in set theory, Bristol, February 2020 | Joel David Hamkins January 22, 2020 Comments for Joel David Hamkins
- Comment on A new proof of the Barwise extension theorem, without infinitary logic by The $Sigma_1$-definable universal finite sequence | Joel David Hamkins January 14, 2020 Comments for Joel David Hamkins
- Comment on Lectures on the philosophy of mathematics, Oxford, Michaelmas term 2019 by Joel David Hamkins January 8, 2020 Comments for Joel David Hamkins
- Comment on Lectures on the philosophy of mathematics, Oxford, Michaelmas term 2019 by Klaus Loehnert January 8, 2020 Comments for Joel David Hamkins
- Comment on Climb into Cantor's attic by Joel David Hamkins January 4, 2020 Comments for Joel David Hamkins