Sakae Fuchino

Reflection Principles formulated as Löwenheim-Skolem Theorems for stationary logics and the Continuum Problem

We give characterizations of variations of Löwenheim-Skolem Theorem for stationary logic (i.e. the logic with monadic second order variables which run over countable subsets of a structure and with the quantifier "there exist stationarily many countable sets such that ...") Löwenheim-Skolem Theorems with reflection cardinal
$<\aleph_2$ for this logic and some variants of it are shown to be equivalent either to the Diagonal Reflection Principle (DRP) down to internally club sets introduced by Sean Cox or this type of DRP plus CH. The Löwenheim-Skolem Theorem for stationary logics with reflection cardinal "$<2^{\aleph_0}$" is not consistent with very large continuum. However, Löwenheim-Skolem Theorem for this logics with reflection cardinal "$<2^{\aleph_0}$" in terms of an internal interpretation of the stationary logic is consistent with the continuum being very large. The Löwenheim-Skolem Theorem for stationary logics with reflection cardinal "$<2^{\aleph_0}$" in terms of stationary logic with an internal ${\cal P}_{\kappa}{\lambda}$-interpretation of second order variables even implies that the continuum is (at least) weakly Mahlo. The results presented in this talk are going to be included in a joint paper with André Ottembreit and Hiroshi Sakai.