Title: Automorphism argument and reverse mathematics
by Keita Yokoyama (Japan Advanced Institute of Science and Technology) as part of Computability theory and applications
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.