Related papers: Forcing Isomorphism II

A forcing extension may create new isomorphisms between two models of a first order theory. Certain model theoretic constraints on the theory and other constraints on the forcing can prevent this pathology. A countable first order theory is…

For which (first-order complete, usually countable) $T$ do there exist non-isomorphic models of $T$ which become isomorphic after forcing with a forcing notion $\mathbb{P}$? Necessarily, $\mathbb{P}$ is non-trivial; i.e.~it adds some new…

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…

We consider the problem of characterizing isomorphisms of types, or, equivalently, constructive cardinality of sets, in the simultaneous presence of disjoint unions, Cartesian products, and exponentials. Mostly relying on results about…

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…

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…

A complete first order theory of a relational signature is called monomorphic iff all its models are monomorphic (i.e. have all the $n$-element substructures isomorphic, for each positive integer $n$). We show that a complete theory…

We show, assuming PD, that every complete finitely axiomatized second order theory with a countable model is categorical, but that there is, assuming again PD, a complete recursively axiomatized second order theory with a countable model…

We introduce a category whose objects are stationary set preserving complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We show that the cut of this category at a rank initial…

The purpose of this paper is to investigate forcing as a tool to construct universal models. In particular, we look at theories of initial segments of the universe and show that any model of a sufficiently rich fragment of those theories…

In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the collections of models of ZFC obtained by Cohen 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).

We answer a question of Moore by building a forcing extension satisfying measuring together with CH. The construction works over any model of ZFC and can be described as a forcing iteration with countable structures as side conditions and…

We note that some form of the condition "$p_1, p_2$ have a $\leq_{\mathbb{Q}}$-lub in $\mathbb{Q}$" is necessary in some forcing axiom for $\lambda$-complete $\mu^+$-c.c. forcing notions. We also show some versions are really stronger than…

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…

We develop a general framework for forcing with coherent adequate sets on $H(\lambda)$ as side conditions, where $\lambda \ge \omega_2$ is a cardinal of uncountable cofinality. We describe a class of forcing posets which we call coherent…

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 null ideal of the…

We present a version with non-definable forcing notions of Shelah's theory of iterated forcing along a template. Our main result, as an application, is that, if $\kappa$ is a measurable cardinal and $\theta<\kappa<\mu<\lambda$ are…

Given a cardinal $\lambda$, category forcing axioms for $\lambda$-suitable classes $\Gamma$ are strong forcing axioms which completely decide the theory of the Chang model $\mathcal C_\lambda$, modulo generic extensions via forcing notions…

We show: There are pairs of universes V_1 subseteq V_2 and there is a notion of forcing P in V_1 such that the change mentioned in the title occurs when going from V_1[G] to V_2[G] for a P-generic filter G over V_2. We use forcing…