Related papers: The exact strength of the class forcing theorem
We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to…
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).
The technique of "classical realizability" is an extension of the method of "forcing"; it permits to extend the Curry-Howard correspondence between proofs and programs, to Zermelo-Fraenkel set theory and to build new models of ZF, called…
I prove forcing preservation theorems for products of definable partial orders preserving the cofinality of the meager or null ideal. Rectangular Ramsey theorems for related ideals follow from the proofs.
Forcing was first introduced by Paul J. Cohen in his work on the independence of the Continuum Hypothesis. Other formulations of forcing appeared using Model Theory, Boolean-valued Models, and Topos Theory. There is a folkloric claim that…
We study principles of the form: if a name $\sigma$ is forced to have a certain property $\varphi$, then there is a ground model filter $g$ such that $\sigma^g$ satisfies $\varphi$. We prove a general correspondence connecting these name…
We consider extensions of the language of Peano arithmetic by transfinitely iterated truth definitions satisfying uniform Tarskian biconditionals. Without further axioms, such theories are known to be conservative extensions of the original…
We develop the theory of partial satisfaction relations for structures that may be proper classes and define a satisfaction predicate appropriate to such structures. We indicate the utility of this theory as a framework for the development…
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…
In the first part of this paper, we consider several natural axioms in urelement set theory, including the Collection Principle, the Reflection Principle, the Dependent Choice scheme and its generalizations, as well as other axioms…
These notes present a compact and self-contained approach to iterated forcing with a particular emphasis on semiproper forcing. We tried to make our presentation accessible to any scholar who has some familiarity with forcing and boolean…
It is shown that the boldface maximality principle for subcomplete forcing, together with the assumption that the universe has only set-many grounds, implies the existence of a (parameter-free) definable well-ordering of…
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 the Zermelo--Fraenkel set theory with the Axiom of Choice a forcing notion is "$\kappa$-distributive" if and only if it is "$\kappa$-sequential". We show that without the Axiom of Choice this equivalence fails, even if we include a weak…
These are lecture notes from a course I gave at the University of Wisconsin during the Spring semester of 1993. Part 1 is concerned with Borel hierarchies. Section 13 contains an unpublished theorem of Fremlin concerning Borel hierarchies…
If T has only countably many complete types, yet has a type of infinite multiplicity then there is a ccc forcing notion Q such that, in any Q --generic extension of the universe, there are non-isomorphic models M_1 and M_2 of T that can be…
We study methods to obtain the consistency of forcing axioms, and particularly higher forcing axioms. We first force over a model with a supercompact cardinal $\theta>\kappa$ to get the consistency of the forcing axiom for $\kappa$-strongly…
Ordinary and transfinite recursion and induction and ZF set theory are used to construct from a fully interpreted object language and from an extra formula a new language. It is fully interpreted under a suitably defined interpretation.…
Extending Mart\'in Escard\'o's effectful forcing technique, we give a new proof of a well-known result: Brouwer's monotone bar theorem holds for any bar that can be realized by a functional of type $(\mathbb{N} \to \mathbb{N}) \to…
The purpose of this article is to give a presentation of the method of forcing aimed at someone with a minimal knowledge of set theory and logic. The emphasis will be on how the method can be used to prove theorems in ZFC.