Abstract We develop technology for investigation of natural forcing extensions of the model $L\left( \mathbb{R} \right)$ which satisfy such statements as “there is an ultrafilter” or “there is a total selector for the Vitali equivalence relation”. The technology reduces many questions about ZF implications between consequences of the Axiom of Choice to natural ZFC forcing problems.

Abstract With everyσ-idealIon a Polish space we associate theσ-idealI* generated by the closed sets inI. We study the forcing notions of Borel sets modulo the respectiveσ-idealsIandI* and find connections between their forcing properties. To this end, we associate to aσ-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 the1–1 or constantproperty ofσ-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: ifIis aσ-ideal generated by closed sets, then either the forcingP1adds a Cohen real, or elseIhas the 1–1 or constant property.

Abstract We present Woodin's proof that if there exists a measurable Woodin cardinal δ then there is a forcing extension satisfying all sentences ϕ such that CH + ϕ holds in a forcing extension of V by a partial order in Vδ. 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 such that M is countably closed in the forcing extension.

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

Abstract 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 ADL(ℝ): roughly ω Woodin cardinals.

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

Abstract It is proved consistent that there be a proper σ-ideal on ω1 and an ℵ1-preserving poset ℙ such that ℙ ⊩ the σ-ideal generated by is not proper.

Abstract We study a generalization of the splitting number s to uncountable cardinals. We prove that 𝔰(κ) > κ+ for a regular uncountable cardinal κ implies the existence of inner models with measurables of high Mitchell order. We prove that the assumption 𝔰(ℵω) > ℵω+1 has a considerable large cardinal strength as well.

Abstract We show that all posets of uniform density ℵ1 may have to add a Cohen real and develop some forcing machinery for obtaining this sort of result.