NOL Seminar with Michael Rathjen
published: 2022-04-11
event date:
2022-04-25
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
Abstract:
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.