Kinna-wagner Selection Principles, Axioms of Choice and Multiple Choice

We study the relationships between weakened forms of the Kinna-Wagner Selection Principle (KW), the Axiom of Choice (AC), and the Axiom of Multiple Choice (MC). Introduction. In this paper we study the relationships between weakened forms of the Kinna-Wagner Selection Principle (KW), the Axiom of Choice (AC), and the Axiom of Multiple Choice (MC). The forms of these statements that we shall use are given below. AC: For every set x of of non-empty sets there is a function f (choice function) such that for each u ∈ x, f(u) ∈ x. MC: For every set x of non-empty sets there is a function f (multiple choice function) such that for each u ∈ x, f(u) is a non-empty finite subset of u. KW: For every set x of sets of cardinality at least 2 there is a function f (KinnaWagner selection function) such that for each u ∈ x, f(u) is a non-empty proper subset of u. We weaken these statements by applying them to well ordered sets. The notation we shall use is as follows: AC: Every well ordered set of non-empty sets has a choice function. ACWO: Every set of non-empty well orderable sets has a choice function. AC WO : Every well ordered set of non-empty well orderable sets has a choice function. AC<א0 : Every set of non-empty finite sets has a choice function. AC <א0 : Every well ordered set of non-empty finite sets has a choice function. ACn: Every set of n-element sets has a choice function. AC n : Every well ordered set of n-element sets has a choice function. 1Eastern Michigan University, Ypsilanti, MI [email protected] 2Brea, CA [email protected] 3Purdue University, West Lafayette, IN [email protected]