--- title: Axiom of Choice description: The Axiom of Choice is a fundamental axiom of set theory concerning the selection of elements from collections of non-empty sets. Learners will understand its equivalence to Zorn's Lemma and the Well-Ordering Theorem, and its implications for modern mathematics. category: mathematics subcategory: set-theory difficulty: beginner, intermediate, advanced url: /subject/axiom-of-choice --- # Axiom of Choice The Axiom of Choice is a fundamental axiom of set theory concerning the selection of elements from collections of non-empty sets. Learners will understand its equivalence to Zorn's Lemma and the Well-Ordering Theorem, and its implications for modern mathematics. ## Available Resources 1 Videos • 2 Courses • 4 Websites ## Websites ### 1. MathWorld: Axiom of Choice Wolfram MathWorld's reference entry on the axiom of choice, giving its formal statement, equivalent forms such as Zorn's lemma and the well-ordering theorem, and its independence from Zermelo-Fraenkel set theory, with citations to the standard literature for further reading. **Difficulty:** Beginner | **Price:** Free **Link:** https://mathworld.wolfram.com/AxiomofChoice.html **Tags:** axiom of choice, set theory, zorn's lemma, well-ordering theorem, zfc ### 2. nLab: Axiom of Choice The nLab entry on the axiom of choice, treating it from a category-theoretic and constructive viewpoint: formulations as every epimorphism splitting, its failure in general toposes, weaker variants like countable and dependent choice, and the Diaconescu theorem linking choice to excluded middle. **Difficulty:** Beginner | **Price:** Free **Link:** https://ncatlab.org/nlab/show/axiom%20of%20choice **Tags:** axiom of choice, category theory, topos theory, constructive mathematics, set theory ### 3. Stanford Encyclopedia of Philosophy: Axiom of Choice **Author:** John L. Bell Peer-reviewed encyclopedia entry by John L. Bell tracing the axiom of choice: its equivalents such as Zorn's lemma and well-ordering, its independence from ZF, constructive objections, and its role in topos theory. Readers learn why the axiom is contested and where mathematics relies on it. **Difficulty:** Intermediate | **Price:** Free **Link:** https://plato.stanford.edu/entries/axiom-choice/ **Tags:** axiom-of-choice, set-theory, zorns-lemma, philosophy-of-mathematics, foundations-of-mathematics ### 4. Wolfram MathWorld **Author:** Eric W. Weisstein 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Videos ### 1. Banach-Tarski Paradox and Axiom of Choice Vsauce video by Michael Stevens walking through the Banach-Tarski paradox, building from countable and uncountable infinities and free groups of rotations to decomposing a sphere into two copies. Viewers see concretely how the axiom of choice permits non-measurable sets and counterintuitive results. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/watch?v=s86-Z-CbaHA **Tags:** banach-tarski-paradox, axiom-of-choice, infinity, non-measurable-sets, set-theory ## Courses ### 1. Introduction to Logic This course is an introduction to Logic from a computational perspective. It shows how to encode information in the form of logical sentences; it shows how to reason with information in this form; and it provides an overview of logic technology and its applications - in mathematics, science, engineering, business, law, and so forth. **Difficulty:** Beginner | **Language:** English | **Duration:** 10 weeks of study, 4-8 hours/week | **Price:** Free **Link:** https://www.coursera.org/learn/logic-introduction **Tags:** courses, mathematics-statistics, discrete-math ### 2. Introduction to Mathematical Thinking **Author:** Keith Devlin Stanford course by Keith Devlin on how mathematicians reason, starting with the logic of language (and, or, not, implication, quantifiers) and moving to the structure of proofs, including contradiction and induction. Learners become able to read, construct and critique simple mathematical proofs. **Difficulty:** Beginner | **Language:** English | **Duration:** Expect to require at least 10 hours of study per week to complete this course satisfactorily. | **Price:** Free **Link:** https://www.coursera.org/learn/mathematical-thinking **Tags:** mathematical-logic, proof-writing, quantifiers, mathematical-reasoning --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/axiom-of-choice