We study an extensive connection between quotient forcings of Borel subsets of Polish spaces modulo a σ-ideal and quotient forcings of subsets of countable sets modulo an ideal.

There is an optimal way of increasing certain cardinal invariants of the continuum.

Let I be an F σ -ideal on natural numbers. We characterize the ultrafilters which are generic over the model L for the poset of I -positive sets of natural numbers ordered by inclusion.

We present two ways in which the model L(R) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing; we show further that a set of ordinals in V cannot be added to L(R) by small forcing. The large cardinal needed corresponds to the consistency strength of AD L (R); roughly ω Woodin cardinals.

We study a generalization of the splitting number s to uncountable cardinals. We prove that $\mathfrak{s}(\kappa) > \kappa^+$ for a regular uncountable cardinal κ implies the existence of inner models with measurables of high Mitchell order. We prove that the assumption $\mathfrak{s}(\aleph_\omega) > \aleph_{\omega + 1}$ has a considerable large cardinal strength as well.

Given a Polish space X and a countable collection of analytic hypergraphs on X, I consider the σ-ideal generated by Borel anticliques for the hypergraphs in the family. It turns out that many of the quotient posets are proper. I investigate the forcing properties of these posets, certain natural operations on them, and prove some related dichotomies.

The cut and choose game is one of the infinitary games on a complete Boolean algebra B introduced by Jech. We prove that existence of a winning strategy for II in implies semiproperness of B. If the existence of a supercompact cardinal is consistent then so is “for every 1-distributive algebra B II has a winning strategy in ”.

We investigate classes of Boolean algebras related to the notion of forcing that adds Cohen reals. A Cohen algebra is a Boolean algebra that is dense in the completion of a free Boolean algebra. We introduce and study generalizations of Cohen algebras: semi-Cohen algebras, pseudo-Cohen algebras and potentially Cohen algebras. These classes of Boolean algebras are closed under completion.

We isolate a forcing which increases the value of δ12 while preserving ω₁ under the assumption that there is a precipitous ideal on ω₁ and a measurable cardinal.

We isolate several large classes of definable proper forcings and show how they include many partial orderings used in practice.

I describe several ways in which forcing arguments can be used to yield clean and conceptual proofs of nonreducibility, ergodicity and other results in the theory of analytic equivalence relations. In particular, I present simple Borel equivalence relationsE, Fsuch that a natural proof of nonreducibility ofEtoFuses the independence of the Singular Cardinal Hypothesis at ℵω.

We show that all posets of uniform density ℵ 1 may have to add a Cohen real and develop some forcing machinery for obtaining this sort of result.

I describe several ways in which forcing arguments can be used to yield clean and conceptual proofs of nonreducibility, ergodicity and other results in the theory of analytic equivalence relations. In particular, I present simple Borel equivalence relations $E, F$ such that a natural proof of nonreducibility of $E$ to $F$ uses the independence of the Singular Cardinal Hypothesis at $\aleph_\omega$.

With every σ-ideal I on a Polish space we associate the σ-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective σ-ideals I and I* and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal. We also study the 1—1 or (...) constant property of σ-ideals, i.e., the property that every Borel function defined on a Borel positive set can be restricted to a positive Borel set on which it either 1—1 or constant. We prove the following dichotomy: if I is a σ-ideal generated by closed sets, then either the forcing P I adds a Cohen real, or else I has the 1—1 or constant property. (shrink)

It is proved consistent that there be a proper σ-ideal ℑ on ω 1 and an ℵ 1 -preserving poset P such that $\mathbb{P} \Vdash$ the σ-ideal generated by ℑ̌ is not proper.

We present Woodin's proof that if there exists a measurable Woodin cardinal δ, then there is a forcing extension satisfying all $\Sigma _{2}^{2}$ sentences ϕ such that CH + ϕ holds in a forcing extension of V by a partial order in V δ . We also use some of the techniques from this proof to show that if there exists a stationary limit of stationary limits of Woodin cardinals, then in a homogeneous forcing extension there is an elementary embedding (...) j: V → M with critical point $\omega _{1}^{V}$ such that M is countably closed in the forcing extension. (shrink)

The relationship between killing ideals and adding reals by forcings is analysed.

Certain separation problems in descriptive set theory correspond to a forcing preservation property, with a fusion type infinite game associated to it. As an application, it is consistent with the axioms of set theory that the circle.

Certain set theoretical notions cannot be split into finer subnotions.

It is proved consistent that there be a proper $\sigma$-ideal $\Im$ on $\omega_1$ and an $\aleph_1$-preserving poset $\mathbb{P}$ such that $\mathbb{P} \Vdash$ the $\sigma$-ideal generated by $\check{\Im}$ is not proper.

In mathematical practice certain formulas φ are believed to essentially decide all other natural properties of the object x. The purpose of this paper is to exactly quantify such a belief for four formulas φ, namely “x is a Ramsey ultrafilter”, “x is a free Souslin tree”, “x is an extendible strong Lusin set” and “x is a good diamond sequence”.

We isolate a forcing which increases the value of $\delta _{2}^{1}$ while preserving ω₁ under the assumption that there is a precipitous ideal on ω₁ and a measurable cardinal.