Boolean valued rings
Boolean-valued selectors for families of sets
Borel and Baire reducibility
We prove that a Borel equivalence relation is classifiable by countable structures if and only if it is Borel reducible to a countable level of the hereditarily countable sets. We also prove the following result which was originally claimed in [FS89]: the zero density ideal of sets of natural numbers is not classifiable by countable structures.
Borel chromatic number of closed graphs
We construct, for each countable ordinal ξ, a closed graph with Borel chromatic number 2 and Baire class ξ chromatic number ℵ₀.
Borel classes in AST. Measurability, cuts and equivalence
Borel completeness of some ℵ₀-stable theories
We study ℵ₀-stable theories, and prove that if T either has eni-DOP or is eni-deep, then its class of countable models is Borel complete. We introduce the notion of λ-Borel completeness and prove that such theories are λ-Borel complete. Using this, we conclude that an ℵ₀-stable theory satisfies for all cardinals λ if and only if T either has eni-DOP or is eni-deep.
Borel equivalence and isomorphism of coanalytic sets [Book]
Borel extensions of Baire measures in ZFC
We prove: 1) Every Baire measure on the Kojman-Shelah Dowker space admits a Borel extension. 2) If the continuum is not real-valued-measurable then every Baire measure on M. E. Rudin's Dowker space admits a Borel extension. Consequently, Balogh's space remains the only candidate to be a ZFC counterexample to the measure extension problem of the three presently known ZFC Dowker spaces.
Borel partitions of unity and lower Carathéodory multifunctions
We prove the existence of Carathéodory selections and representations of a closed convex valued, lower Carathéodory multifunction from a set A in into a separable Banach space Y, where ℰ is a sub-σ-field of the Borel σ-field ℬ(E) of a Polish space E, X is a Polish space and A is the Suslin operation. As applications we obtain random versions of results on extensions of continuous functions and fixed points of multifunctions. Such results are useful in the study of random differential equations...
Borel sets of exact class
Borel sets with -section
Borel sets with large squares
For a cardinal μ we give a sufficient condition (involving ranks measuring existence of independent sets) for: if a Borel set B ⊆ ℝ × ℝ contains a μ-square (i.e. a set of the form A × A with |A| =μ) then it contains a -square and even a perfect square, and also for if has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute” way. Assuming for transparency, those three conditions (, and ) are equivalent, and from this we deduce that...
Borel Tukey morphisms and combinatorial cardinal invariants of the continuum
We discuss the Borel Tukey ordering on cardinal invariants of the continuum. We observe that this ordering makes sense for a larger class of cardinals than has previously been considered. We then provide a Borel version of a large portion of van Douwen's diagram. For instance, although the usual proof of the inequality 𝔭 ≤ 𝔟 does not provide a Borel Tukey map, we show that in fact there is one. Afterwards, we revisit a result of Mildenberger concerning a generalization of the unsplitting and splitting...
Boréliens à coupes
Borel-Wadge degrees
Two sets of reals are Borel equivalent if one is the Borel pre-image of the other, and a Borel-Wadge degree is a collection of pairwise Borel equivalent subsets of ℝ. In this note we investigate the structure of Borel-Wadge degrees under the assumption of the Axiom of Determinacy.
Bounded dually residuated lattice ordered monoids as a generalization of fuzzy structures
Bounded lattices with antitone involutions and properties of MV-algebras
We introduce a bounded lattice L = (L;∧,∨,0,1), where for each p ∈ L there exists an antitone involution on the interval [p,1]. We show that there exists a binary operation · on L such that L is term equivalent to an algebra A(L) = (L;·,0) (the assigned algebra to L) and we characterize A(L) by simple axioms similar to that of Abbott's implication algebra. We define new operations ⊕ and ¬ on A(L) which satisfy some of the axioms of MV-algebra. Finally we show what properties must be satisfied by...
Bounded products in the theory of valued fields.
Bounded sets in topological vector spaces.
Bounded statements in the theory of algebraically closed fields with distinguished automophisms.