Some combinatorial theorems equivalent to the prime ideal theorem

Abstract

Abstract.Some useful combinatorial selection lemmas are shown to be directly equivalent to the prime ideal theorem for boolean algebras.1. The theorems we shall consider are intimately related to R. Rado's selection lemma (Theorem 2, below) which first appeared in [10] and subsequently has found wide application (see [1], [3], [4], [12]). Our main theorem is Theorem 1 which we use to derive other forms of Rado's lemma and to prove A. Robinson's valuation lemma which was shown by Robinson in [9] to be a fundamental result in model theory.We also show that Theorem 1 and some of the theorems we derive from it are equivalent to the prime ideal theorem for boolean algebras and thus we have some useful abstract versions of this theorem, versions divorced from any particular algebraic structure. Whether the original lemma of Rado is as strong as the prime ideal theorem appears to be an open question; note, however, that E. S. Wölk [12] has shown that Rado's lemma plus the axiom of choice for families of finite sets implies the Tychonoff theorem for finite spaces, which, in turn, implies the prime ideal theorem (see [8]). We conjecture that Rado's lemma is strictly weaker than the prime ideal theorem. Preliminaries.In this paper we shall work, informally, within the framework of Zermelo-Fraenkel set theory (ZF). All uses of the axiom of choice will be explicitly noted.Given a set /, by a partial function on I, we mean a function whose domain is a subset of /. A partial valuation on I is a partial function on / whose range is included in {0, 1}. As in ZF we consider functions to be special sets of ordered pairs and we even allow 0 , the empty function (the empty set of ordered pairs). If/ is a partial function on / we write D(f) for its domain, and if t/e/ we write/ft/ for the restriction of/to UnD(f).

Variants of RADO'S Selection Lemma and their Applications

“…Condition ( 5 ) guarantees that Bp g p , and finally, we conclude from condition (6) that R -p is multiplicatively closed.…”

“…Ve lmve seen that the Ultrafilter Principle implies RADO'S Lemma. As to the converse, it has been conjectured by COWEN[6] that RADO'S Lemma is strictly weaker than the Ultrafilter Principle and we second this conjecture. (See the observation following Corollary 2.3.)…”

“…The first generalizes a coinbinatori;tl theorem of COWEN [6] and RICE [27]. I\-e supplement the results of COWEN [6] and sliow in a self contained way that the various selection lemmns, except RADO'S lemma, are all equivalent in ZF set theory to the Ultrafilter Principle. The arguments in this and in the subsequent sections can be formalized in ZF set theory, and of course the Axiom of Choice is nowhere assumed.…”

“…Furtherinore, it is known [9] that the Ultrafilter Principle is independent of the axioms of ZERMELO-FRAENKEL (ZF) set theory (without the Axiom of Choice). Different versions of RADO'S lernma were subsequently and independently discovered by LoS and RYLL-NARDZEWSKI [ 171, ENGELER [7], and ROBINSON [SS].I n Section 2 of this paper two dtiitional selection lemmas are introduced.The first generalizes a coinbinatori;tl theorem of COWEN [6] and RICE [27]. I\-e supplement the results of COWEN [6] and sliow in a self contained way that the various selection lemmns, except RADO'S lemma, are all equivalent in ZF set theory to the Ultrafilter Principle.…”

“…Different versions of RADO'S lernma were subsequently and independently discovered by LoS and RYLL-NARDZEWSKI [ 171, ENGELER [7], and ROBINSON [SS].I n Section 2 of this paper two dtiitional selection lemmas are introduced.The first generalizes a coinbinatori;tl theorem of COWEN [6] and RICE [27]. I\-e supplement the results of COWEN [6] and sliow in a self contained way that the various selection lemmns, except RADO'S lemma, are all equivalent in ZF set theory to the Ultrafilter Principle. The arguments in this and in the subsequent sections can be formalized in ZF set theory, and of course the Axiom of Choice is nowhere assumed.I n Section 3 we study the strength of RADO'S lemma.…”