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

- On checking a proof July 24, 2017 Dave Sixsmith
- Second-order transfinite recursion is equivalent to Kelley-Morse set theory over GBC July 23, 2017 Joel David Hamkins
- Nebula-The cryptocurrency that will produce the reversible computer July 22, 2017 Joseph Van Name
- The transitive multiverse July 22, 2017 Asaf Karagila
- The fundamental problem of math on the web July 21, 2017 Peter Krautzberger

### Comments on Boole’s Rings

- Comment on Infinite Combinatorial Topology by Rodrigo HernÃ¡ndez-GutiÃ©rrez July 25, 2017 Comments for Assaf Rinot
- Comment on Open determinacy for class games by Second-order transfinite recursion is equivalent to Kelley-Morse set theory over GBC | Joel David Hamkins July 23, 2017 Comments for Joel David Hamkins
- Comment on Transfinite recursion as a fundamental principle in set theory by Second-order transfinite recursion is equivalent to Kelley-Morse set theory over GBC | Joel David Hamkins July 23, 2017 Comments for Joel David Hamkins
- Comment on Games with the computable-play paradox by Warren D Smith July 21, 2017 Comments for Joel David Hamkins
- Comment on Ordinal definable subsets of singular cardinals by saf July 18, 2017 Comments for Assaf Rinot