Alexander B. Kharazishvili, Set Theoretical Aspects of Real Analysis, CRC Press, Taylor & Francis Group, Boca Raton 2015, xxii + 433 pp, ISBN: 13: 978-1-4822- 4201-0. - ANISIU, Valeriu (Studia Universitatis Babeș-Bolyai Mathematica, 2015)
As it is well known the Axiom of Choice (AC) plays a fundamental role inmathematics, many fundamental results depending on it, or even being equivalentto it. At the same time the acceptance of AC, i.e. working within ZFC (Zermelo-Frenkel set theory + AC), leads to counterintuitive and paradoxical results, the mostintriguing being the Banach-Tarski paradox. On the other side, the restriction toZF axioms also leads to paradoxical situations as, for instance, the possibility torepresent the set R of real numbers as a countable union of countable sets (see page11 of the book). For these reasons the mathematicians tried to understand to whatextent some results depend essentially on AC, or on some weaker variants: CountableChoice (CC), Dependent Choice (DC), Product Countable Choice (PCC), ContinuumHypothesis (CH), Martin's Axiom (MA). Some of the results equivalent to AC are:(1) Tychonov's theorem on the compactness of the product of compact spaces; (2)Kuratowski-Zorn lemma on the existence of maximal chains in partially ordered sets;(3) the total ordering of cardinal numbers; (3) the existence of a Hamel basis in anyvector space. Other results, as (i) the existence of nonmeasurable subsets of R; (ii)the existence of subsets of R not having the Baire property; (iii) the existence of aHamel basis in R (considered as a vector space over Q); the Hahn-Banach theorem,cannot be proved within ZF and need some uncountable forms of the AC. As it isknown, Solovay constructed a mathematical theory based on ZF plus the existence ofa nonmeasurable Lebesgue set, taken as an axiom.