Paper Info
Title
Constructive Order Theory
Abstract
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.
Year
DOI
Venue
2001
10.1002/1521-3870(200105)47:2<211::AID-MALQ211>3.0.CO;2-U
MATHEMATICAL LOGIC QUARTERLY
Keywords
Field
DocType
Boolean lattice,axiom of choice,axiom of multiple choice,chain,constructive,contraction,directed set,inductive set,supremum,well-ordered set
Discrete mathematics,Set theory,Combinatorics,Zermelo–Fraenkel set theory,Algebra,Constructive,Order theory,Morse–Kelley set theory,Urelement,Constructive set theory,Well-order,Mathematics
Journal
Volume
Issue
ISSN
47
2
0942-5616
Citations 
PageRank 
References 
0
0.34
0
Authors
1
Name
Order
Citations
PageRank
Marcel Erné12910.77