We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the meager ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler’s classical result in the theory of forcing.

We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the meager ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler’s classical result in the theory of forcing.

Our aim is to present a version of short extenders forcing which splits between cardinals above and below the critical cardinal and allows to blow up the power arbitrary higher. 1 The Main Preparation Forcing The definition of the forcing below follows the ideas of [1], [2], [3], [4]. We extend the method of [4] to gaps up to κ. The forcing adds a certain kind of a simplified morass with linear...

Our aim is to present a version of short extenders forcing which splits between cardinals above and below the critical cardinal and allows to blow up the power arbitrary higher. 1 The Main Preparation Forcing The definition of the forcing below follows the ideas of [1], [2], [3], [4]. We extend the method of [4] to gaps up to κ. The forcing adds a certain kind of a simplified morass with linear...

We investigate the consistency strength of the forcing axiom for Σ3 formulas, for various classes of forcings. We review that the consistency strength of Σ3-absoluteness for all set forcing or even just for ω1-preserving forcing is that of a reflecting cardinal. To get the same strength from the forcing axiom restricted to proper forcing, one can add the hypotheses that ω1 is inaccessible to re...

Prikry asked if it is relatively consistent with the usual axioms of ZFC that every nontrivial ccc forcing adds either a Cohen or a random real. Both Cohen and random reals have the property that they neither contain nor are disjoint from an infinite set of integers in the ground model, i.e. they are splitting reals. In this note I show that that it is relatively consistent with ZFC that every ...

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.

The forcing polynomial and anti-forcing are two important enumerative polynomials associated with all perfect matchings of a graph. In graph large order, the exhaustive enumeration which is used to compute number given matching too time-consuming number. this paper, we come up an efficient method — integer linear programming, matching. As applications, obtain di-forcing C60 , C70 C72 as consequ...