Details
| Original language | English |
|---|---|
| Pages (from-to) | 513-536 |
| Number of pages | 24 |
| Journal | Commentationes Mathematicae Universitatis Carolinae |
| Volume | 38 |
| Issue number | 3 |
| Publication status | Published - 1997 |
Abstract
We study several choice principles for systems of finite character and prove their equivalence to the Prime Ideal Theorem in ZF set theory without Axiom of Choice, among them the Intersection Lemma (stating that if S is a system of finite character then so is the system of all collections of finite subsets of ∪S meeting a common member of S), the Finite Cutset Lemma (a finitary version of the Teichmüller-Tukey Lemma), and various compactness theorems. Several implications between these statements remain valid in ZF even if the underlying set is fixed. Some fundamental algebraic and order-theoretical facts like the Artin-Schreier Theorem on the orderability of real fields, the Erdös-De Bruijn Theorem on the colorability of infinite graphs, and Dilworth's Theorem on chain-decompositions for posets of finite width, are easy consequences of the Intersection Lemma or of the Finite Cutset Lemma.
Keywords
- Axiom of choice, Compact, Consistent, Prime ideal, Subbase, System of finite character
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Commentationes Mathematicae Universitatis Carolinae, Vol. 38, No. 3, 1997, p. 513-536.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Prime Ideal Theorems and systems of finite character
AU - Erné, Marcel
PY - 1997
Y1 - 1997
N2 - We study several choice principles for systems of finite character and prove their equivalence to the Prime Ideal Theorem in ZF set theory without Axiom of Choice, among them the Intersection Lemma (stating that if S is a system of finite character then so is the system of all collections of finite subsets of ∪S meeting a common member of S), the Finite Cutset Lemma (a finitary version of the Teichmüller-Tukey Lemma), and various compactness theorems. Several implications between these statements remain valid in ZF even if the underlying set is fixed. Some fundamental algebraic and order-theoretical facts like the Artin-Schreier Theorem on the orderability of real fields, the Erdös-De Bruijn Theorem on the colorability of infinite graphs, and Dilworth's Theorem on chain-decompositions for posets of finite width, are easy consequences of the Intersection Lemma or of the Finite Cutset Lemma.
AB - We study several choice principles for systems of finite character and prove their equivalence to the Prime Ideal Theorem in ZF set theory without Axiom of Choice, among them the Intersection Lemma (stating that if S is a system of finite character then so is the system of all collections of finite subsets of ∪S meeting a common member of S), the Finite Cutset Lemma (a finitary version of the Teichmüller-Tukey Lemma), and various compactness theorems. Several implications between these statements remain valid in ZF even if the underlying set is fixed. Some fundamental algebraic and order-theoretical facts like the Artin-Schreier Theorem on the orderability of real fields, the Erdös-De Bruijn Theorem on the colorability of infinite graphs, and Dilworth's Theorem on chain-decompositions for posets of finite width, are easy consequences of the Intersection Lemma or of the Finite Cutset Lemma.
KW - Axiom of choice
KW - Compact
KW - Consistent
KW - Prime ideal
KW - Subbase
KW - System of finite character
UR - http://www.scopus.com/inward/record.url?scp=33750711925&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:33750711925
VL - 38
SP - 513
EP - 536
JO - Commentationes Mathematicae Universitatis Carolinae
JF - Commentationes Mathematicae Universitatis Carolinae
SN - 0010-2628
IS - 3
ER -