### A classification of definable forcings on ω1

Under the assumption of the existence of sharps for reals all simply definable posets on ${\omega}_{1}$ are classified up to forcing equivalence.

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Under the assumption of the existence of sharps for reals all simply definable posets on ${\omega}_{1}$ are classified up to forcing equivalence.

We compare the forcing-related properties of a complete Boolean algebra $\mathbb{B}$ with the properties of the convergences ${\lambda}_{\mathrm{s}}$ (the algebraic convergence) and ${\lambda}_{\mathrm{ls}}$ on $\mathbb{B}$ generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that ${\lambda}_{\mathrm{ls}}$ is a topological convergence iff forcing by $\mathbb{B}$ does not produce new reals and that ${\lambda}_{\mathrm{ls}}$ is weakly topological if $\mathbb{B}$ satisfies condition $\left(\hslash \right)$ (implied by the $\U0001d531$-cc). On the other hand, if ${\lambda}_{\mathrm{ls}}$ is a weakly topological convergence, then $\mathbb{B}$ is a ${2}^{\U0001d525}$-cc algebra...

Assuming large cardinals, we show that every κ-complete filter can be generically extended to a V-ultrafilter with well-founded ultrapower. We then apply this to answer a question of Abe.

We present a proof of the Boolean Prime Ideal Theorem in a transitive model of ZF in which the Axiom of Choice does not hold. We omit the argument based on the full Halpern-Läuchli partition theorem and instead we reduce the proof to its elementary case.

A new ⋄-like principle ${\diamond}_{}$ consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that $\neg {\diamond}_{}$ is consistent with CH and that in many models of = ω₁ the principle ${\diamond}_{}$ holds. As ${\diamond}_{}$ implies that there is a MAD family of size ℵ₁ this provides a partial answer to a question of J. Roitman who asked whether = ω₁ implies = ω₁. It is proved that ${\diamond}_{}$ holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether = ω₁ in such models....

In A theorem on supports in the theory of semisets [Comment. Math. Univ. Carolinae 14 (1973), no. 1, 1–6] B. Balcar showed that if $\sigma \subseteq D\in M$ is a support, $M$ being an inner model of ZFC, and $\mathcal{P}(D\setminus \sigma )\cap M=r``\sigma $ with $r\in M$, then $r$ determines a preorder "$\u2aaf$" of $D$ such that $\sigma $ becomes a filter on $(D,\u2aaf)$ generic over $M$. We show that if the relation $r$ is replaced by a function $\mathcal{P}(D\setminus \sigma )\cap M={f}_{-1}\left(\sigma \right)$, then there exists an equivalence relation "$\sim $" on $D$ and a partial order on $D/\sim \phantom{\rule{0.166667em}{0ex}}$ such that $D/\sim \phantom{\rule{0.166667em}{0ex}}$ is a complete Boolean algebra, $\sigma /\sim \phantom{\rule{0.166667em}{0ex}}$ is a generic filter and ${\left[f\left(u\right)\right]}_{\sim}=-\sum (u/\sim )$ for any $u\subseteq D$, $u\in M$.

We present two ${\mathbb{P}}_{max}$ varations which create maximal models relative to certain counterexamples to Martin’s Axiom, in hope of separating certain classical statements which fall between MA and Suslin’s Hypothesis. One of these models is taken from [19], in which we maximize relative to the existence of a certain type of Suslin tree, and then force with that tree. In the resulting model, all Aronszajn trees are special and Knaster’s forcing axiom ₃ fails. Of particular interest is the still open question...

We prove that if there exists a Cohen real over a model, then the family of perfect sets coded in the model has a disjoint refinement by perfect sets.

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 ω₂ using finite conditions.