σ-homomorphisms from the Borel σ-algebra into the Loeb σ-algebra

σ-ring and σ-algebra of Sets1

In this article, semi-ring and semi-algebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semi-ring of sets has already been formalized in [12], that is, strictly speaking, a definition of a quasi semi-ring of sets suggested in the last few decades [14]. We adopt a classical definition of a semi-ring of sets here to avoid such a confusion. Ring ...

Borel homomorphisms of smooth σ-ideals

Given a countable Borel equivalence relation E on a Polish space, let IE denote the σ-ideal generated by the Borel partial transversals of E. We show that there is a Borel homomorphism from IE to IF if and only if there is a smooth-to-one Borel homomorphism from a finite index Borel subequivalence relation of E to F . As a corollary, we see that IE is homogeneous in the sense of Zapletal (2007,...

The Extender Algebra and Vagaries of Σ 21 Absoluteness Philipp

We review the construction of the extender algebra, a Boolean algebra which is due to Woodin, with δ-many generators. The resulting genericity iteration is applied to prove a new Σ1-absoluteness theorem for c.c.c. forcings with ordinal parameters. Additionally we introduce and discuss sets that extend to a class with unique condensation. We analyse the sets that extend to classes with unique co...

The Extender Algebra and Vagaries of Σ 21 Absoluteness

We review the construction of the extender algebra, a Boolean algebra which is due to Woodin, with δ-many generators. The resulting genericity iteration is applied to prove a new Σ1-absoluteness theorem for c.c.c. forcings with ordinal parameters. We also introduce and discuss sets of ordinals that extend to a class with unique condensation. This paper mostly deals with the extender algebra, a ...

σ-Normality of Topological Spaces