The idea of “forcing” has long been used in many research fields, such as colorings, orientations, geodetics and dominating sets in graph theory, as well as Latin squares, block designs and Steiner systems in combinatorics (see [1] and the references therein). Recently, the forcing on perfect matchings has been attracting more researchers attention. A forcing set of M is a subset of M contained...

the idea of “forcing” has long been used in many research fields, such as colorings, orientations, geodetics and dominating sets in graph theory, as well as latin squares, block designs and steiner systems in combinatorics (see [1] and the references therein). recently, the forcing on perfect matchings has been attracting more researchers attention. a forcing set of m is a subset of m contained...

The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a condition in the relevant generic filter. We show that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma ca...

In this paper we study the notion of forcing for Lukasiewicz predicate logic ( L∀, for short), along the lines of Robinson’s forcing in classical model theory. We deal with both finite and infinite forcing. As regard to the former we prove a Generic Model Theorem for L∀, while for the latter, we study the generic and existentially complete standard models of L∀.

If κ < λ are such that κ is a strong cardinal whose strongness is indestructible under κ-strategically closed forcing and λ is weakly compact, then we show that A = {δ < κ | δ is a non-weakly compact Mahlo cardinal which reflects stationary sets} must be unbounded in κ. This phenomenon, however, need not occur in a universe with relatively few large cardinals. In particular, we show how to cons...

We present a technique for coding sets “into K,” where K is the core model below a strong cardinal. Specifically, we show that if there is no inner model with a strong cardinal then any X ⊂ ω1 can be made ∆3 (in the codes) in a reasonable and stationary preserving set generic extension.

We use model theoretic forcing to study and generalize the construction of (K ,≤)-generic models introduced by Kueker and Laskowski. We characterize the (K ,≤)-generic models in terms of forcing and introduce a more general class of models, called essential forcing generics, which have many of the same properties.

We introduce and study the first-order Generic Vopěnka’s Principle, which states that for every definable proper class of structures C of the same type, there exist B 6= A in C such that B elementarily embeds into A in some set-forcing extension. We show that, for n ≥ 1, the Generic Vopěnka’s Principle fragment for Πn-definable classes is equiconsistent with a proper class of n-remarkable cardi...

We present some results about generics for computable Mathias forcing. The n-generics and weak n-generics in this setting form a strict hierarchy as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if G is any n-generic with n ≥ 3 then it satisfies the jump property G(n−1) = G′ ⊕ ∅(n). We prove that every such G has generalized high degree, ...

Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Levy-Collapse. These show in particular that certain applications of forcing axioms require to add generic countable sequences high up in the set-theoretic hierarchy even before collapsing everything down to א1. Later we give applications, among them the c...