Details
| Originalsprache | Englisch |
|---|---|
| Seiten (von - bis) | 211-222 |
| Seitenumfang | 12 |
| Fachzeitschrift | Mathematical logic quarterly |
| Jahrgang | 47 |
| Ausgabenummer | 2 |
| Publikationsstatus | Veröffentlicht - 1 März 2001 |
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.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Logik
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in: Mathematical logic quarterly, Jahrgang 47, Nr. 2, 01.03.2001, S. 211-222.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Constructive order theory
AU - Erné, Marcel
PY - 2001/3/1
Y1 - 2001/3/1
N2 - 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.
AB - 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.
KW - Axiom of choice
KW - Axiom of multiple choice
KW - Boolean lattice
KW - Chain
KW - Constructive
KW - Contraction
KW - Directed set
KW - Inductive set
KW - Supremum
KW - Well-ordered set
UR - http://www.scopus.com/inward/record.url?scp=0035531235&partnerID=8YFLogxK
U2 - 10.1002/1521-3870(200105)47:2<211::AID-MALQ211>3.0.CO;2-U
DO - 10.1002/1521-3870(200105)47:2<211::AID-MALQ211>3.0.CO;2-U
M3 - Article
AN - SCOPUS:0035531235
VL - 47
SP - 211
EP - 222
JO - Mathematical logic quarterly
JF - Mathematical logic quarterly
SN - 0942-5616
IS - 2
ER -