On -hereditary Sets and Consequences of the Axiom of Choice

Abstract

We will prove that some so-called union theorems (see [2]) are equivalent in ZF 0 to statements about the transitive closure of relations. The special case of "bounded" union theorems dealing with κ-hereditary sets yields equivalents to statements about the transitive closure of κ-narrow relations. The instance κ = ω1 (i. e., hereditarily countable sets) yields an equivalent to Howard-Rubin's Form 172 (the transitive closure Tc(x) of every hereditarily countable set x is countable). In particular, the countable union theorem (Howard-Rubin's Form 31) and, a fortiori, the axiom of countable choice imply Form 172. Mathematics Subject Classification: 03E25, 03E99.We will work in one of the familiar systems of set theory without the axiom of choice (AC) and the axiom of foundation, e. g., in ZF 0 . We use customary set-theoretic notation and notions, 2) and though all theorems of this paper can be reformulated in "pure" set theory, we will also make use of syntactical devices such as (informal) class variables and class terms. Throughout this paper infinite cardinals κ, λ, . . . will always be well-ordered cardinals, i. e., alephs; κ + is the cardinal successor of cardinal κ.For the reader's convenience, we mention here some further particular notions. For any two classes A and B, A \ B denotes the set-theoretical difference. Given an arbitrary relation R, i. e., a class of ordered pairs a, b , R −1 is the converse relation, R | X is the restriction of R to class X, i. e., R | X := R ∩ (X × X), and R X is the domain-restriction of R to X. R[X] denotes the R-image of class X, fd(R) = dom(R) ∪ rng(R) is the field of R, and R • S is the relative product of relations R and S. For any two functions f and g, f · g denotes the composition, defined 1) e-mail: Servitor@mi.uni-koeln.de 2) For the reader's convenience we have tried to stick to Howard-Rubin's terminology as close as possible without giving up the terminology and notation of [1]. There is, however, a difference that should be mentioned here. Our "ZF 0 " is just Zermelo-Fraenkel set theory without the axiom of foundation, (see, e. g., Lévy [4], Vaught [5]) whereas Howard-Rubin's "ZF 0 " admits atoms, but requires foundation for sets of sets.