Chains, Antichains and Fences

Abstract

Chains and antichains are arguably the most common kinds of ordered sets in mathematics. The elementary number systems ℕ , ℤ, ℚ and ℝ (with the exception of course being ℂ) are chains. Chains are also at the heart of set theory. The Axiom of Choice is equivalent to Zorn's Lemma (which we will adopt as an axiom) and the Well-Ordering Theorem. Both latter results are results about chains.