Prime Ideal Theorems and systems of finite character

authored by

Marcel Erné

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.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

Commentationes Mathematicae Universitatis Carolinae

Volume

38

Pages

513-536

No. of pages

24

ISSN

0010-2628

Publication date

1997

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

General Mathematics