Automorphism argument and reverse mathematics

Abstract: In the study of models of Peano (or first-order) arithmetic, there are many results on recursively saturated models and their automorphisms. Here, we apply such an argument to models of second-order arithmetic and see that any countable recursively saturated model (M,S) of WKL_0* is isomorphic to its countable coded omega-submodel if Sigma_1-induction fails in (M,S). From this result, we see some interesting but weird properties of WKL_0* with the absence of Sigma_1-induction such as the collapse of analytic hierarchy. This argument can also be applied to the reverse mathematical study of Ramsey's theorem for pairs (RT22), and we see some new relations between the computability-theoretic characterizations of RT22 and the famous open question on the first-order part of RT22+RCA_0. This work is a part of a larger project joint with Marta Fiori Carones, Leszek Kolodziejczyk, Katarzyna Kowalik and Tin Lok Wong.

logic

Audience: researchers in the topic

Computability theory and applications

Series comments: Description: Computability theory, logic

The goal of this endeavor is to run a seminar on the platform Zoom on a weekly basis, perhaps with alternating time slots each of which covers at least three out of four of Europe, North America, Asia, and New Zealand/Australia. While the meetings are always scheduled for Tuesdays, the timezone varies, so please refer to the calendar on the website for details about individual seminars.