A Note on Applications of the Löwenheim‐Skolem‐Theorem in General Topology

Abstract

The idea of the proof of the Lowenheim-Skolem-Theorem is applied in many fields of mathematics. For instance, this type of argument is called the "closure argument" (see JUHASZ [S]) in the theory of cardinal functions in topology and plays an important role. The set-theoretic version of the Lowenheim-Skolem-Theorem says that for any set there are (countable) elementary substructures. I ngives a striking application of elementary substructures, proving a decomposition theorem for arbitrary partially ordered sets. Another application can be found in SHELAH [14]. Nevertheless, this method has not yet been recognized in general topology, although as method of proof "elementary substructures " are in general elegant and simple in representation. Once the concept of elementary substructures has been understood, it can be used as a convenient tool for studying difficult problems. The aim of this paper is to discuss some rather simple applications of elementary substructures.Set theory here means always Zermelo-Fraenkel set theory including the axiom of choice. Let H be an arbitrary non-empty set. If ip(x1, . . . , x,,) is a formula of the language of set theory with the only free variables x, , . . . , x,,, then H C ip[a,, . . . , a,,] denotes that the relativization of ip to H is satisfied by interpretation of the variables x1 , . . . , x,, by the elements a , , . . ., a,, E H . Formally, H P ip[a,, . . . , a,,] may be defined by induction on ip (cf. e.g. KUNEN [9]). More briefly, the relativization of ip is obtained from q~ by replacing all quantifiers 32, Vx by (32 E H ) , (Vx E H ) , respectively. If M is a nonempty subset of H , we say that M is an elementary substructure of H ( M < H , for short) if for every formula ip(xl, . . ., 2,) and for every a , , . . ., a,, E M we have M k-ip[a, , . . . , a,,] 0 H I = ip[a,, . . . , a,,]. A frequently used argument is the following: If M < H and ip(x, , . . . , x,, , x) is a given formula, such that for a , , . . ., a,, E M there exists an a E H , such that H P ip[a,, . . . , a,,, a], then there is a b E M with H k-ip[a,, . . . , a,,, b] (and therefore M P ~[ a , , . . .,a,,, b]). If for a , , . . ., a,, E M there exists a unique a E H satisfying H C ip[a, , . . . , a,, , a], then a belongs to M ; in this case we say that a is definable from a,, . . . , a,, by the formula ip or, if x is the only free variable of ip, a is defined by ip. I n the sequel, as usually, the role of H is played by the set HB of all sets hereditarily of cardinality <O, where 8 is a sufficiently large regular cardinal. For regular uncountable 8, H, is a model of all axioms of set theory, with possible exception of the power set axiom. A : 1 B I < A ] , where IBI denotes the cardinality of B. By 1 t (A, t cardinals) we denote the fact that A < t and a@ < t for all LY < t and For any set A and any cardinal A we set [A]<i = {B < A.