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Constructive Order Theory
Author(s) -
Erné Marcel
Publication year - 2001
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 - 10.1002/1521-3870(200105)47:2<211::aid-malq211>3.0.co;2-u
Subject(s) - mathematics , axiom of choice , constructive set theory , constructive , power set , infimum and supremum , axiom , zermelo–fraenkel set theory , partially ordered set , joins , discrete mathematics , urelement , statement (logic) , closure (psychology) , set theory , constructive proof , set (abstract data type) , finite set , equivalence (formal languages) , computer science , process (computing) , programming language , mathematical analysis , geometry , political science , economics , law , market economy , operating system
We introduce the notion of constructive suprema and of constructively directed sets. The Axiom of Choice turns out to be equivalent to the postulate that every supremum is constructive, but also to the hypothesis that every directed set admits a function assigning to each finite subset an upper bound. The Axiom of Multiple Choice (which is known to be weaker than the full Axiom of Choice in set theory without foundation) implies a simple set‐theoretical induction principle (SIP), stating that any system of sets that is closed under unions of well‐ordered subsystems and contains all finite subsets of a given set must also contain that set itself. This is not provable without choice principles but equivalent to the statement that the existence of joins for constructively directed subsets of a poset follows from the existence of joins for nonempty well‐ordered subsets. Moreover, we establish the equivalence of SIP with several other fundamental statements concerning inductivity, compactness, algebraic closure systems, and the exchange between chains and directed sets.