Borel classes in AST. Measurability, cuts and equivalence. (English) Zbl 0676.03032

Kalina, Martin; Zlatoš, Pavol

Commentat. Math. Univ. Carol. 30, No. 2, 357-372 (1989).

Borel classes create the least \(\sigma\)-ring containing the ring of set- theoretically definable classes (set parameters are admitted in definitions). There is a close relationship between these classes and classical Borel sets. It is obvious that in alternative set theory we do not need any topology for their definition as it is in the classical case. The authors investigate them from the point of view of their magnitude given by a Loeb-type measure. They also examine the equivalence and subvalence relations determined by Borel mappings and Borel cuts which are connected with the considered matter. E.g., they prove the Cantor-Bernstein theorem for Borel mappings. The paper is elaborated in a nice and exhaustive manner.

Reviewer: M.Čuda

MSC:

03E70	Nonclassical and second-order set theories
03H99	Nonstandard models
28E05	Nonstandard measure theory
03E15	Descriptive set theory
28A05	Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets

Keywords:

measure; real class; lower cut; upper cut; Borel equivalence; Borel cardinal; Borel classes; alternative set theory; Cantor-Bernstein theorem