We make a formal analogy between random sampling and fresh name generation. We show that quasi-Borel spaces, a model for probabilistic programming, can soundly interpret the ν-calculus, a calculus for name generation. Moreover, we prove that this semantics is fully abstract up to first-order types. This is surprising for an ‘off-the-shelf’ model, and requires a novel analysis of probability distributions on function spaces. Our tools are diverse and include descriptive set theory and normal forms for the ν-calculus.

We study the spectrum of forcing notions between the iterations of σ-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of α-proper forcings for indecomposable countable ordinals α, the Axiom A forcings and forcings completely embeddable into an iteration of a σ-closed followed by a ccc forcing. For the latter class, we present an equivalent characterization in terms of Baumgartnerʼs Axiom A. This resolves a conjecture of Baumgartner from the 1980s. We also study the bounded forcing axioms for the hierarchy of α-proper forcings. Following ideas of Shelah we separate them for distinct countable indecomposable ordinals.

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.