...common with each member of x. (Two of the best known equivalent formulations are: Zorn's Lemma; if each chain in a partially ordered set has an upper bound, then there is a maximal element of the set, and The Well-Ordering Principle; each set can be well-ordered.) The continuum hypothesis is: (CH) Each...

...chain. The language is suggestive, and pictures should help keep the concepts in mind. Lemma 1.34 (Zorn) If every chain in a partially ordered set has an upper bound, then the set has a maximal element. This assertion is, in fact, equivalent to the axiom of choice. We shall...

...every system of sets. (b) (The Well-Ordering Principle) Every set can be well-ordered. (c) (Zorn's Lemma) If every chain in a partially ordered set has an upper bound. then the partially ordered set has a maximal element. We remind the reader that a chain is a linearly ordered...

...of Choice". One of them is the so-called "Zorn's Lemma", which we state next. THEOREM 3.1.6 (Zorn's Lemma) If every chain in a partially ordered set has...bound, then there is a maximal element of the set. REMARK 3.1.7 Throughout this book we always assume the ZermeloFrankel set theory (ZF) together with...