相关论文: Universal forcing notions and ideals

We develop a toolbox for forcing over arbitrary models of set theory without the axiom of choice. In particular, we introduce a variant of the countable chain condition and prove an iteration theorem that applies to many classical forcings…

A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of "for every suitable forcing…

We deal with an iteration theorem of forcing notion with a kind of countable support of nice enough forcing notion which is proper aleph_2-c.c. forcing notions. We then look at some special cases (Q_D 's preceded by random forcing).

Let $\mathcal{N}$ be the $\sigma$-ideal of the null sets of reals. We introduce a new property of forcing notions that enable control of the additivity of $\mathcal{N}$ after finite support iterations. This is applied to answer some open…

We study the spectrum of forcing notions between the iterations of $\sigma$-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of $\alpha$-proper forcings for indecomposable countable ordinals as well as…

We present a sufficient condition for irreducibility of forcing algebras and study the (non)-reducedness phenomenon. Furthermore, we prove a criterion for normality for forcing algebras over a polynomial base ring with coefficients in a…

We show that for a Suslin ccc forcing notion $\mathbb Q$ adding a Hechler real, ``$\text{ZF}+\text{DC}_{\omega_1}+$all sets of reals are $I_{\mathbb Q,\aleph_0}$-measurable'' implies the existence of an inner model with a measurable…

With every $\sigma$-ideal $I$ on a Polish space we associate the $\sigma$-ideal $I^*$ generated by the closed sets in $I$. We study the forcing notions of Borel sets modulo the respective $\sigma$-ideals $I$ and $I^*$ and find connections…

We give a brief survey on the interplay between forcing axioms and various other non-constructive principles widely used in many fields of abstract mathematics, such as the axiom of choice and Baire's category theorem. First of all we…

Using a countable support product of creature forcing posets, we show that consistently, for uncountably many different functions the associated Yorioka ideals' uniformity numbers can be pairwise different. In addition we show that, in the…

We describe a method of building ``nice'' sigma-ideals from Souslin ccc forcing notions. [These notes were written down in 1992, but were not submitted to any journal. In a slightly modified form, they were incorporated to: T. Bartoszynski…

We show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $\omega_2$ in forcing extensions by the pure side condition forcing introduced by Neeman. The…

We look for a parallel to the notion of ``proper forcing'' among lambda-complete forcing notions not collapsing lambda^+ . We suggest such a definition and prove that it is preserved by suitable iterations.

In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of the generalised Cantor space and study its associated ideal of null sets and notion of measurability. This issue was…

The Steprans forcing notion arises as a quotient of Borel sets modulo the ideal of $\sigma$-continuity of a certain Borel not $\sigma$-continuous function. We give a characterization of this forcing in the language of trees and using this…

The foundations of forcing theory are reworked to streamline the presentation and to show how the most basic results are applicable in very general contexts.

In this paper we consider the Foreman's maximality principle, which says that any non-trivial forcing notion either adds a new real or collapses some cardinals. We prove the consistency of some of its consequences. We prove that it is…

We deal with (< kappa)-supported iterated forcing notions which are (E_0,E_1)-complete, have in mind problems on Whitehead groups, uniformizations and the general problem. We deal mainly with the successor of a singular case. This continues…

We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a…

We investigate how set-theoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for information about a model of set theory $\langle M,\in^M\rangle$, we explain senses in which one may compute…