100 Notre Dame Journal of Formal Logic Volume 34, Number 1, Winter 1993 The Strength of the Δ-system Lemma PAUL HOWARD and JEFFREY SOLSKI Abstract The delta system lemma is not provable in set theory without the axiom of choice nor does it imply the axiom of choice. / Introduction A Δ system 8 is a collection of sets such that there is a set r with the property that (VA E S) (v£ G S) (A Φ B => A ΠB = r). r is called the root of S The A system lemma is the statement: ASL For every uncountable collection T of finite sets there is an uncount able subcollection 8 of T which forms a A system. ASL is provable in Zermelo Fraenkel set theory (ZF) with the axiom of choice (AC) as shown by Kunen [3], [4]. We will investigate the strength of ASL in ZF (without the axiom of choice). In this theory there are two possible defi nitions of Xis uncountable: \X\ φ K o or K o < \X\. These definitions are equiv alent if AC is assumed. In Section 2 below we will use the first definition exclusively. In Section 3 we will investigate the consequences of using the sec ond definition. We will also refine ASL in the following way: ASL(n) will denote, for each positive integer w, the Δ system lemma for families of AZ element sets. We note that ASL(l) is trivially true. Our main goal will be to prove that for any inte ger n > 2, ASL(n) is equivalent to ASL and also to the conjunction of the two statements: CU The union of a countable collection of countable sets is countable. and PC Every uncountable collection of countable sets has an uncountable sub collection with a choice function. 2 Using the first definition of uncountable We begin with: Lemma 2.1 ZFV (yn E ω {0})(ASL(n + 1) => ASL(n)). Received June 24, 1991; revised January 4, 1993