--- title: Continuum Hypothesis description: The Continuum Hypothesis proposes that there is no set whose cardinality is strictly between that of the integers and the real numbers. Learners will understand its independence from Zermelo-Fraenkel set theory and its historical significance in mathematical logic. category: mathematics subcategory: set-theory difficulty: beginner, intermediate, advanced url: /subject/continuum-hypothesis --- # Continuum Hypothesis The Continuum Hypothesis proposes that there is no set whose cardinality is strictly between that of the integers and the real numbers. Learners will understand its independence from Zermelo-Fraenkel set theory and its historical significance in mathematical logic. ## Available Resources 1 Videos • 2 Books • 2 Courses • 3 Websites ## Websites ### 1. Joel David Hamkins Blog: Continuum Hypothesis Joel David Hamkins's research blog posts on the continuum hypothesis, including the set-theoretic multiverse view, forcing arguments, and whether CH has a definite answer. Readers see how a working set theorist reasons about independence results and the philosophical debate they provoke. **Difficulty:** Beginner | **Price:** Free **Link:** https://jdh.hamkins.org/tag/continuum-hypothesis/ **Tags:** continuum-hypothesis, set-theory, set-theoretic-multiverse, forcing, philosophy-of-mathematics ### 2. Wikipedia: Continuum Hypothesis Encyclopedia article on Cantor's continuum hypothesis, covering cardinality, Gödel's and Cohen's independence proofs, the generalized continuum hypothesis, and current views among set theorists, with extensive references. Readers get a map of the problem's history and the key results needed for further study. **Difficulty:** Beginner | **Price:** Free **Link:** https://en.wikipedia.org/wiki/Continuum_hypothesis **Tags:** continuum-hypothesis, set-theory, cardinality, independence-results ### 3. 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 ## Courses ### 1. Set Theory Learn Set Theory & the Continuum Hypothesis with Joel David Hamkins' video course! Explore key concepts and expand your mathematical understanding. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PL5KkMZvBpo5AH_5GpxMiryJT6Dkj32H6N **Tags:** set-theory, continuum-hypothesis, forcing, cardinal-arithmetic, mathematical-logic ### 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 ## Videos ### 1. How To Count Past Infinity A Vsauce video by Michael Stevens explaining ordinal and cardinal numbers, from counting with omega to the distinction between countable and uncountable sets. Viewers come away understanding why some infinities are larger than others and how transfinite counting works. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/watch?v=SrU9YDoXE88 **Tags:** ordinal-numbers, cardinal-numbers, uncountable-sets, infinity ## Books ### 1. Set Theory (Studies in Logic, Vol. 34) **Author:** Kenneth Kunen Graduate treatment of what ZFC can and cannot prove. Formalises the axioms, builds the von Neumann hierarchy, absoluteness and Godel's constructible universe, then develops forcing to establish the independence of the continuum hypothesis and the axiom of choice. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/1848900503?tag=edmonddante07-20 **Tags:** set-theory, zfc, forcing, independence-proofs, constructible-universe, continuum-hypothesis ### 2. Set Theory and the Continuum Hypothesis **Author:** Paul J. Cohen Paul Cohen's monograph presenting his proof that the continuum hypothesis and the axiom of choice are independent of standard set theory, introducing the forcing method, with background on logic and Gödel's constructible universe. Readers follow the original argument from its inventor. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0486469212?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, set-theory --- *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/continuum-hypothesis