Banach-Tarski Paradox and Axiom of Choice
Vsauce
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.
More resources on Axiom of Choice
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.
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.
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.
Stanford Encyclopedia of Philosophy: Axiom of Choice
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.
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.
Introduction to Mathematical Thinking
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.