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 classifiable if it is superstable and does not have either the dimensional order property or the omitting types order property. Shelah [Sh:c] showed that if a theory T is classifiable then each model of cardinality lambda is described by a...

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. This technique will be used to suggest a unified perspective on generic absoluteness principles.

Suppose that $P$ is a forcing notion, $L$ is a language (in $V$), $\dot{\tau}$ a $P$-name such that $P\Vdash$ "$\dot{\tau}$ is a countable $L$-structure". In the product $P\times P$, there are names $\dot{\tau_{1}},\dot{\tau_{2}}$ such that for any generic filter $G=G_{1}\times G_{2}$ over $P\times P$, $\dot{\tau}_{1}[G]=\dot{\tau}[G_{1}]$ and $\dot{\tau}_{2}[G]=\dot{\tau}[G_{2}]$. Zapletal asked whether or not $P \times P \Vdash \dot{\tau}_{1}\cong\dot{\tau}_{2}$ implies tha...

This is an expository paper about several sophisticated forcing techniques closely related to standard finite support iterations of ccc partial orders. We focus on the four topics of ultrapowers of forcing notions, iterations along templates, Boolean ultrapowers of forcing notions, and restrictions of forcing notions to elementary submodels.

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 from $\Gamma$. $\mathsf{MM}^{+++}$ was the first category forcing axiom to be isolated (by the second author). In this paper we present, without proofs, a general theory of category forcings, and prove the existence of $\aleph_1$-many pairw...

In the present paper we are interested in simple forcing notions and Forcing Axioms. A starting point for our investigations was the article [JR1] in which several problems were posed. We answer some of those problems here.

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 others.

We find new "reasons" for a class of models for not having a universal model in a cardinal $\lambda$. This work, though it has consequences in model theory, is really in combinatorial set theory. We concentrate on a prototypical class which is a simply defined class of models, of combinatorial character - models of $T_{\rm ceq}$ (essentially another representation of $T_{\rm feq}$ which was already considered but the proof with $T_{\rm ceq}$ is more transparent). Models of $T...

In this article we adapt the existing account of class-forcing over a ZFC model to a model $(M,\mathcal{C})$ of Morse-Kelley class theory. We give a rigorous definition of class-forcing in such a model and show that the Definability Lemma (and the Truth Lemma) can be proven without restricting the notion of forcing. Furthermore we show under which conditions the axioms are preserved. We conclude by proving that Laver's Theorem does not hold for class-forcings.

The main purpose of this note is to present a matrix version of Neeman's forcing with sequences of models of two types. In some sense, the forcing defined here can be seen as a generalization of Asper\'o-Mota's pure side condition forcing from arXiv:1203.1235 and arXiv:1206.6724. In our case, the forcing will consist of certain symmetric systems of countable and uncountable elementary submodels of some $H(\kappa)$, which will be closed by the relevant intersections, in Neeman...