We define a forcing poset which adds a club subset of a given fat stationary set S ⊆ ω2 with finite conditions, using S-adequate sets of models as side conditions. This construction, together with the general amalgamation results concerning S-adequate sets on which it is based, is substantially shorter and simpler than our original version in [3]. The theory of adequate sets introduced in [2] p...

We continue the study of adequate sets which we began in [2] by introducing the idea of a strongly adequate set, which has an additional requirement on the overlap of two models past their comparison point. We present a forcing poset for adding a club to a fat stationary subset of ω2 with finite conditions, thereby showing that a version of the forcing posets of Friedman [1] and Mitchell [3] fo...

In [4] a preservation theorem for countable support iterated forcing is proved with restriction to forcing notions which are not ω-distributive. We give the proof of the theorem without this restriction. 1. The preservation theorem. In [4] a preservation theorem for countable support iteration of proper forcing notions was proved with the additional assumption that all forcing notions which are...

A natural generalization of Cohen's set of forcing conditions (the t w ~ valued functions with domain a finite subset of w) is the set of twovalued functions with domain an element of an ideal J on ~. The I~roblem treated in this paper is to determine when such forcing yields a generic real of minimal degree of constructibility. A :;imple decomposition argument shows that the non-maximality of ...

If ≤ is a preorder on a set P and p0 ≤ p1, we say that p1 is an extension of p0. Recall that a preorder is separative if and only if whenever p1 is not an extension of p0 there is an extension of p1 which is incompatible with p0. We say that P = (P,≤) is a forcing notion (also forcing poset) if ≤ is a separative preorder with minimal element 0P. Note that if P is separative and p1 p̌0 ∈ Ġ then p...

Using the notion of anti fuzzy points and its besideness to and nonquasi-coincidence with a fuzzy set, new concepts in anti fuzzy subalgebras in BCK/BCI-algebras are introduced and their properties and relationships are investigated.

This chapter describes, following the historical development, the investigation of large cardinal hypotheses using the method of forcing. Large cardinal hypotheses, also regarded as strong axioms of infinity, have stimulated a vast mainstream of modern set theory, and William Mitchell’s chapter in this volume deals with their investigation through inner models, Menachem Kojman’s chapter with th...

Using a theorem from pcf theory, we show that for any singular cardinal ν, the product of the Cohen forcing notions on κ, κ < ν adds a generic for the Cohen forcing notion on ν. The following question (problem 5.1 in Miller’s list [Mi91]) is attributed to Rene David and Sy Friedman: Does the product of the forcing notions n2 add a generic for the forcing ω+12? We show here that the answer is ye...

Let G be a graph that admits a perfect matching. A forcing set for a perfect matching M of G is a subset S of M , such that S is contained in no other perfect matching of G. This notion has arisen in the study of .nding resonance structures of a given molecule in chemistry. Similar concepts have been studied for block designs and graph colorings under the name de/ning set, and for Latin squares...