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The Axiom of Choice in Second‐Order Predicate Logic
Author(s) -
Gaßner Christine
Publication year - 1994
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.19940400410
Subject(s) - unary operation , axiom of choice , mathematics , trichotomy (philosophy) , zermelo–fraenkel set theory , disjoint sets , constructive set theory , discrete mathematics , urelement , predicate (mathematical logic) , first order logic , axiom , binary relation , choice function , binary number , combinatorics , set theory , set (abstract data type) , arithmetic , computer science , epistemology , philosophy , geometry , programming language
Abstract The present article deals with the power of the axiom of choice (AC) within the second‐order predicate logic. We investigate the relationship between several variants of AC and some other statements, known as equivalent to AC within the set theory of Zermelo and Fraenkel with atoms, in Henkin models of the one‐sorted second‐order predicate logic with identity without operation variables. The construction of models follows the ideas of Fraenkel and Mostowski. It is e. g. shown that the well‐ordering theorem for unary predicates is independent from AC for binary predicates and from the trichotomy law for unary predicates. Moreover, we show that the AC for binary predicates follows neither from the trichotomy law for unary predicates nor from Zorn's lemma for unary predicates nor from the formalization of the axiom of choice for disjoint families of sets for binary predicates, and that the trichotomy law for unary predicates does not follow from AC for binary predicates. Mathematics Subject Classification: 03B15, 03E25, 04A25.