Related papers: On $\omega_3$-chains in P($\omega_1$) mod finite

We force over the constructible universe to obtain a model of the $\Pi^1_3$-reduction property, thus lowering the best known large cardinal strength from the existence of $M_1^{\#}$ to just ZFC. In this model the $\Pi^1_3$-uniformization…

We consider strong combinatorial principles for sigma-directed families of countable sets in the ordering by inclusion modulo finite, e.g. P-ideals of countable sets. We try for principles as strong as possible while remaining compatible…

Any $n$-vertex $3$-graph with minimum codegree at least $\lfloor n/3\rfloor$ must have a spanning tight component, but immediately below this threshold it is possible for no tight component to span more than $\lceil 2n/3\rceil$ vertices.…

We show that higher Sacks forcing at a regular limit cardinal and club Miller forcing at an uncountable regular cardinal both add a diamond sequence. We answer the longstanding question, whether $\kappa = \kappa^{<\kappa} \geq\aleph_1$…

We give a combinatorial characterization of countable submaximal subspaces of $2^\kappa$. Using a parametrized version of Mathias forcing, we prove that there exists a countable submaximal subspace of $2^{\omega_1}$ whilst…

We introduce the idea of a coherent adequate set of models, which can be used as side conditions in forcing. As an application we define a forcing poset which adds a square sequence on $\omega_2$ using finite conditions.

We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model $M$ of $\mathit{ZFC}$, of generic extensions satisfying $\mathit{ZFC}+\neg\mathit{CH}$ and $\mathit{ZFC}+\mathit{CH}$.…

Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles…

The idea of this paper is to explore the existence of canonical countably saturated models for different classes of structures. It is well-known that, under CH, there exists a unique countably saturated linear order of cardinality…

An $\omega_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $\omega_1$-compact space is $\sigma$-countably compact, i.e., the union of…

We answer a question of Woodin by showing that assuming an inaccessible cardinal $\kappa$ which is a limit of ${<}\kappa$-supercompact cardinals exists, there is a stationary set preserving forcing $\mathbb{P}$ so that $V^{\mathbb…

We give an exposition of an iteration theorem for iterating $(<\lambda)$-closed stationary $\lambda^+$-cc forcing with supports of size $<\lambda$ and preserving these two properties. We discuss the relation of this theorem with other…

Assume $\kappa = \kappa^{< \kappa}$ (usually $\aleph_0$ or an inaccessible). We shall deal with iterated forcings preserving ${}^{\kappa>}{\rm Ord}$ and not collapsing cardinals along a linear order $L$. A sufficient condition for this,…

This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are two parts. In the first half I consider alternative versions of the Cicho\'n diagram. First I…

We try to control many cardinal characteristics by working with a notion of orthogonality between two families of forcings. We show that b^+<g is consistent

We construct, from a ground model of $ZFC$, a transitive symmetric model $M$ satisfying $ZF + DC + PP + AC_{wo} + \neg AC$. The construction starts with a Cohen symmetric seed model $N$ over $Add(\omega,\omega_1)$ and performs an Ord-length…

We consider the modality "$\varphi$ is true in every $\sigma$-centered forcing extension", denoted $\square\varphi$, and its dual "$\varphi$ is true in some $\sigma$-centered forcing extension", denoted $\lozenge\varphi$ (where $\varphi$ is…

We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of…

In the present paper we shall prove that countable \omega-categorical simple CM-trivial theories and countable \omega-categorical simple theories with strong stable forking are low. In addition, we observe that simple theories of bounded…

We introduce a class of notions of forcing which we call $\Sigma$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $\Sigma$-Prikry. We show that given…