Related papers: Universal classes near $\aleph_1$

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework…

We prove that a random group, in Gromov's density model with $d<1/16$, satisfies a universal sentence $\sigma$ (in the language of groups) if and only if $\sigma$ is true in a nonabelian free group.

In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal $\kappa \ge \omega_2$, $\textsf{ISP}(\kappa)$ implies that $\textsf{SCH}$ holds above $\kappa$, and (3) forcing posets…

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 define and investigate HC-forcing invariant formulas of set theory, whose interpretations in the hereditarily countable sets are well behaved under forcing extensions. This leads naturally to a notion of cardinality ||Phi|| for sentences…

We provide a short proof of Shelah's eventual categoricity conjecture, assuming the Generalized Continuum Hypothesis ($GCH$), for abstract elementary classes (AEC's) with interpolation, a strengthening of amalgamation which is a necessary…

Let K be an Abstract Elemenetary Class satisfying the amalgamation and the joint embedding property, let \mu be the Hanf number of K. Suppose K is tame. MAIN COROLLARY: (ZFC) If K is categorical in a successor cardinal bigger than…

We define a new class of infinitary logics $\mathscr L^1_{\kappa,\alpha}$ generalizing Shelah's logic $\mathbb L^1_\kappa$ defined in \cite{MR2869022}. If $\kappa=\beth_\kappa$ and $\alpha <\kappa$ is infinite then our logic coincides with…

We prove that a random group, in Gromov's density model with $d < 1/16$ satisfies with overwhelming probability a universal-existential first-order sentence $\sigma$ (in the language of groups) if and only if $\sigma$ is true in a…

In [FHK13], the authors considered the question whether model-existence of $L_{\omega_1,\omega}$-sentences is absolute for transitive models of ZFC, in the sense that if $V \subseteq W$ are transitive models of ZFC with the same ordinals,…

We show that for any uncountable cardinal $\lambda$, the category of sets of cardinality at least $\lambda$ and monomorphisms between them cannot appear as the category of point of a topos, in particular is not the category of models of a…

Answering some of the main questions from [MR13], we show that whenever $\kappa$ is a cardinal satisfying $\kappa^{< \kappa} = \kappa > \omega$, then the embeddability relation between $\kappa$-sized structures is strongly invariantly…

This paper provides two extensions of first order logic by `$\omega$-rules'. In each case we characterize the countable structures whose theory in the logic is categorical (has a unique model). In the one-sorted inferential $\omega$-logic,…

Theorem: There is a {\em complete sentence} $\phi$ of $L_{\omega_1,\omega}$ such that $\phi$ has maximal models in a set of cardinals $\lambda$ that is cofinal in the first measurable $\mu$ while $\phi$ has no maximal models in any $\chi…

We show that if \kappa\ is a weakly compact cardinal then the embeddability relation on (generalized) trees of size \kappa\ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space 2^\kappa\…

We derive a new sufficient condition for the existence of {\omega}-categorical universal structures in classes of relational structures with constraints, augmenting results by Cherlin, Shelah, Chi, and Hubi\v{c}ka and Ne\v{s}et\v{r}il.…

Set-theoretic axioms formulated in terms of existence of a Laver-generic large cardinal were introduced in [16] and studied further in [17], [18], [20]. These axioms, let us call them Laver-genericity axioms, claim the existence of a…

We give Woodin's original proof that if there exists a $(\kappa+2)-$strong cardinal $\kappa,$ then there is a generic extension of the universe in which $\kappa=\aleph_\omega,$ $GCH$ holds below $\aleph_\omega$ and…

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…

For many classes of models, there are universal members in any cardinal $\lambda$ which "essentially satisfies GCH", i.e. $\lambda = 2^{< \lambda}$, in particular for the class of a complete first order $T$ (well, if at least $\lambda >…