Zorn's Lemma and Some Applications
by Keith Conrad · University of Connecticut
Expository notes stating Zorn's lemma and applying it to construct maximal ideals, bases of vector spaces, algebraic closures, and other objects across group theory, ring theory, linear algebra, and topology, with a closing discussion of its non-constructive character.
More resources on Well-Ordering Theorem
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
ProofWiki: Well-Ordering Theorem
ProofWiki page stating Zermelo's theorem that every set can be well-ordered, with proofs from the axiom of choice via a choice function, transfinite induction and transfinite recursion, each step linked to its underlying definitions and lemmas. Two of the proofs are marked incomplete.
nLab: Well-ordering theorem
nLab entry on the theorem that every set admits a well-ordering: Zermelo's 1904 proof from the axiom of choice, its use in defining cardinals as initial ordinals, and recent results on its status in constructive mathematics, with primary-source references.
The Axiom of Choice
The definitive ceiling for this topic — the only source here that treats the well-ordering theorem inside the full landscape of choice, including what fails without it. Graduate monograph on the axiom of choice: its equivalents including the well-ordering theorem and Zorn's lemma, consistency and independence via permutation models and forcing, embedding theorems, and systematic study of weaker choice principles and their consequences.
245B Notes 7: Well-ordered sets, ordinals, and Zorn's lemma
Graduate course notes developing well-ordered sets, order isomorphism, ordinals, and transfinite induction, then proving the well-ordering principle and Zorn's lemma from the axiom of choice. Includes exercises and commentary on why the constructions feel non-constructive.
Part II — Logic and Set Theory (Cambridge Lecture Notes)
Full typeset notes for Cambridge's Part II Logic and Set Theory course. Covers well-orderings, ordinals, Hartogs' lemma, transfinite induction, posets, Zorn's lemma, and the well-ordering principle with complete proofs, plus propositional and predicate logic and ZF.