The Scandinavian Logic Society

NOL Seminar with Michael Rathjen

The Nordic Online Logic Seminar is organised monthly over Zoom, with expository talks on topics of interest for the broader logic community. The seminar is open for professional or aspiring logicians and logic aficionados worldwide. If you wish to receive the Zoom ID and password for it, as well as further announcements, please subscribe here: https://listserv.gu.se/sympa/subscribe/nordiclogic.

Next talk: Monday, 25 April, 16.00-17.30 CEST (UTC+2), on Zoom (details are provided to the seminar subscribers)

Title: Completeness: Turing, Schütte, Feferman (and Löb)

Speaker: Michael Rathjen, Professor in Pure Mathematics, University of Leeds

Progressions of theories along paths through Kleene’s Omega adding the consistency of the previous theory at every successor step, can deduce every true Π10-statement. This was shown by Turing in his 1938 thesis who called these progressions “ordinal logics”. In 1962 Feferman proved the amazing theorem that progressions based on the “uniform reflection principle” can deduce every true arithmetic statement. In contrast to Turing’s, Feferman’s proof is very complicated, involving several cunning applications of self-reference via the recursion theorem. Using Schütte’s method of search trees (or decomposition trees) for omega-logic and reflexive induction, however, one can give a rather transparent proof.