An infinitary-logic-free proof of the Barwise end-extension theorem, with new applications, University of Münster, January 2019

This will be a talk for the Logic Oberseminar at the University of Münster, January 11, 2019.

Abstract. I shall present a new proof, with new applications, of the amazing extension theorem of Barwise (1971), which shows that every countable model of ZF has an end-extension to a model of ZFC + V=L. This theorem is both (i) a technical culmination of Barwise’s pioneering methods in admissible set theory and the admissible cover, but also (ii) one of those rare mathematical results saturated with significance for the philosophy of set theory. The new proof uses only classical methods of descriptive set theory, and makes no mention of infinitary logic. The results are directly connected with recent advances on the universal $\Sigma_1$-definable finite set, a set-theoretic version of Woodin’s universal algorithm.

The Logic Bike

In 2004 I was a Mercator Gastprofessor at Universität Münster, Institut für mathematische Logik  und Grundlagenforschung, where I was involved with interesting mathematics, particularly with Ralf Schindler and Gunter Fuchs, who is now at CUNY.

At that time, I had bought a bicycle, and celebrated Münster’s incredible bicycle culture, a city where the number of registered bicycles significantly exceeds the number of inhabitants.  I have long thought that Münster gets something fundamentally right about how to live in a city with bicycles, and the rest of the world should take note.  I am pleased to say that in recent years, New York City is becoming far more bicycle-friendly, although we don’t hold a candle to Münster.

At the end of my position, I donated the bicycle to the Logic Institute, where it has now become known as the Logic Bike, and where I have recently learned that over the years it has now been ridden by a large number of prominent set theorists; it must be one of the few bicycles in the world to have its own web page!