A Study of the Interrelation between Fuzzy Topological Systems and Logics

Abstract

diagrams of the countable-models of Kripke-Platek set theory (KP). We show that, given a path f through TKP, representing a model M of KP, and another computable ill-founded-branching tree T , if f fails to compute a path through T , then M assigns to T a nonstandard ordinal tree rank. Furthermore, we indicate some circumstances in which, given computable-branching trees T 0 and T 1 , a fixed path through TKP helps the paths through T 0 compute paths through T 1. In a different line of work, we consider effective forcing notions. In particular, we define a class of effective forcing notions that are similar to versions of Mathias forcing and Cohen forcing defined in the literature, and prove some results about how these notions relate. As a consequence, we see that the generics for an effective version of Mathias forcing compute generics for an effective version of Hechler forcing, and vice-versa. Later, we focus on a notion of Mathias forcing over a countable Turing ideal, defined by Cholak, Dzhafarov, and Soskova. We show that there are nested Turing ideals for which the Mathias generics for the larger ideal do not all compute Mathias generics for the smaller ideal.