Set-ΛΛαppings on Dedekind Sets

HajnaPs free set principle is equivalent to the axiom of choice, and some of its variants for Dedekind-finite sets are equivalent to countable forms of the axiom of choice. As was observed by Freiling [4] in the case of X = IR, if one applies some heuristics about selecting elements of X at random, one obtains assertions of the form "Every set-mapping/:X-> I (x£f(x)) has a nontrivial free subset H (i.e., x£f(y) for {x,y} E [Z/])", where /is some ideal in P(X) (e.g., /=Lebesgue null sets or / = countable subset). Here we consider the ideal / = [X], the well-orderable subsets of X, and we show that some "randomness" axioms with respect to this ideal are equivalent to variants of the axiom of choice AC. (The use of the term "randomness" in this context is justified since, for some sets X, /indeed is an ideal of null sets for some measure; cf. [6], p. 148.) Theorem 1 In ZF, the following assertions are equivalent: (ϊ)AC, (ii) For every set-mapping f: X-+ [X] there is a co-well-orderable free set H (Le.,X\HG [X])> (iii) For every set-mapping f: K -» [κ] < λ , λ < |κ| fλ a well-orderable cardinal number, \κ\ the not necessarily well-orderable Scott cardinal number of K), there is a free set H of cardinality \ K |, (iv) If S: X~ -• P{E) is a ramification system, then for each g E E there is an f which is maximal (with respect to inclusion) in {h E X~:g E S(h)}. Proof: That AC implies (ii) is immediate from the well-ordering theorem; a proof of HajnaPs theorem (iii) is in [3], p. 276, and of the ramification lemma (iv) in [3], p. 83. (ii) => (i) and (iii) => (i): Let θ be the Hartogs-number of | κ\ (the cardinality of X). The function/:/c X Θ-+ [K X θ] defined byf(x,a) = [x] X a is a set-mapping such that the cardinality \HΠ ({x} x θ)\ < 1 for all x E K, if His Received October 14, 1986; revised April 9, 1987 and July 2, 1987