Two Classical Surprises Concerning the Axiom of Choice and the Continuum Hypothesis

1. INTRODUCTION. In this paper we introduce the reader to two remarkable results in the theory of sets. Both are more than fifty years old, but neither one appears to be well known among nonspecialists. Each one states that a certain proposition implies the Axiom of Choice. First we describe the results, then review definitions, then, finally, present the proofs, most of which are straightforward. Our first surprise concerns Trichotomy, which states that any two (infinite) cardinals a and b are comparable-i.e., either a < b or a = b or a > b. In the absence of special assumptions, Trichotomy may not be taken for granted. In fact, Friedrich Hartogs proved in 1915 that Trichotomy implies the Axiom of Choice. (This is the surprise.) It is an astounding result, as Trichotomy appears to be an isolated proposition. In addition , the same paper of Hartogs makes a crucial contribution to the "continuum problem ," which is to decide where c, the cardinal of the continuum (i.e., the cardinal of the