An incompleteness theorem for βn models

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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AMS/ASL Special Session on Alan Turing

I arrived home yesterday from the 2012 Joint Mathematics Meetings in Boston, where I was a co-organizer of the AMS/ASL Special Session on the Life and Legacy of Alan Turing. The talks were wonderful, and the session went very smoothly, for which I can thank my co-organizers Jeff Hirst and Damir Dzhafarov. Continue reading

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The logic of Reverse Mathematics

This post is about a research idea I have been thinking about which is quite different from my usual research. It’s an example of a project with an “old fashioned” feel to it, as if it could have been studied 50 years ago. It’s almost a toy problem, so I haven’t spent too long digging through references yet. For all I know it was solved 50 years ago. Continue reading

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Moving to WordPress

For a while I have been thinking about migrating to a content management system for my web page. Previously I had a script I wrote that was essentially a simply wiki to let me edit pages from any web browser (which is important to me). This worked, but it was somewhat primitive.

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