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Continuum hypothesis

Definition

In mathematics, specifically set theory, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states:There is no set whose cardinality is strictly between that of the integers and the real numbers.

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Abraham Fraenkel Absolute infinite Abstract logic Ackermann set theory Aleph number Algebraic logic Almost Alphabet (formal languages)Alternative set theory Amorphous set Argument Arity Atomic formula Atomic model (mathematical logic)Atomic sentence Automata theory Automated theorem proving Axiom Axiom of adjunction Axiom of choice Axiom of constructibility Axiom of countable choice Axiom of dependent choice Axiom of determinacy Axiom of extensionality Axiom of global choice Axiom of infinity Axiom of limitation of size Axiom of pairing Axiom of power set Axiom of projective determinacy Axiom of regularity Axiom of union Axiom schema Axiom schema of replacement Axiom schema of specification Axiomatic set theory Axiomatic system Axiomatization of Boolean algebras Banach–Tarski paradox Beppo Levi Bertrand Russell Beth number Bijection Binary operation Boolean algebra Boolean algebras canonically defined Boolean function Burali-Forti paradox Cantor's diagonal argument Cantor's first uncountability proof Cantor's paradox Cantor's theorem Cardinal number Cardinality Cardinality of the continuum Cartesian product Categorical theory Category (mathematics)Category of sets Category theory Chapman & Hall Church encoding Church–Turing thesis Class (set theory)Classical logic Codomain Cofinality Compactness theorem Complement (set theory)Complete theory Computability theory Computable function Computable set Computably enumerable set Concrete category Conjecture Conservative extension Consistency Consistent Constructible universe Construction of the real numbers Constructive set theory Continuum (set theory)Continuum assumption Countable set David E. 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Hilbert's eighteenth problem Hilbert's eighth problem Hilbert's eleventh problem Hilbert's fifteenth problem Hilbert's fifth problem Hilbert's first problem Hilbert's fourteenth problem Hilbert's fourth problem Hilbert's nineteenth problem Hilbert's ninth problem Hilbert's problems Hilbert's second problem Hilbert's seventeenth problem Hilbert's seventh problem Hilbert's sixteenth problem Hilbert's sixth problem Hilbert's tenth problem Hilbert's third problem Hilbert's thirteenth problem Hilbert's twelfth problem Hilbert's twentieth problem Hilbert's twenty-first problem Hilbert's twenty-fourth problem Hilbert's twenty-second problem Hilbert's twenty-third problem Hilbert system History of logic History of mathematical logic Image (mathematics)Inaccessible cardinal Independence (mathematical logic)Inference Infinite-valued logic Infinite set Information theory Inhabited set Injective function Inner model Integer International Congress of Mathematicians Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Intuitionistic logic Isomorphism Joel David Hamkins John von Neumann Journal für die Reine und Angewandte Mathematik Journal of Symbolic Logic Kenneth Kunen Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Kurt Gödel Kőnig's theorem (set theory)Lambda calculus Large cardinal Large cardinal axiom Lemma (mathematics)Lindström's theorem List of Hilbert systems List of axioms List of first-order theories List of formal systems List of mathematical theories List of set identities and relations List of statements independent of ZFC Logic Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical truth Logicism Löwenheim–Skolem theorem Many-valued logic Map (mathematics)Martin's axiom Martin's maximum Material conditional MathWorld Mathematical analysis Mathematical logic Mathematical 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Propositional calculus Propositional formula Propositional variable Quantifier (logic)Quantifier rank Rational Number Rational number Real number Recursion Recursive set Relation (mathematics)Reverse mathematics Richard Dedekind Robinson arithmetic Rule of inference Russell's paradox Saharon Shelah Satisfiability Saturated model Schröder–Bernstein theorem Second-order arithmetic Second-order logic Second continuum hypothesis Self-verifying theories Semantic theory of truth Semantics of logic Sentence (mathematical logic)Sequent calculus Set-builder notation Set (mathematics)Set theory Signature (logic)Singleton (mathematics)Skolem Skolem's paradox Skolem arithmetic Solomon Feferman Soundness Spectrum of a sentence Spectrum of a theory Square of opposition Strength (mathematical logic)String (formal languages)Structure (mathematical logic)Subset Substitution (logic)Substructure (mathematics)Supertask Surjective function Suslin's problem Syllogism Symbol (formal)Symmetric difference Syntax (logic)T-schema Tarski's axiomatization of the reals Tarski's axioms Tarski's theory of truth Tarski's undefinability theorem Tarski–Grothendieck set theory Tautology (logic)Term (logic)Term logic The Continuum Hypothesis (album)Theorem Theories of truth Theory (mathematical logic)Thomas Jech Thoralf Skolem Three-valued logic Timeline of mathematical logic Topology Transfer principle Transfinite induction Transitive set True arithmetic Truth predicate Truth table Truth value Tuple Turing machine Type (model theory)Type theory Ultrafilter (set theory)Ultrafilter on a set Ultraproduct Uncountable set Undecidable problem Uninterpreted function Union (set theory)Uniqueness quantification Universal quantification Universal set Universe (mathematics)Urelement Validity (logic)Variable (mathematics)Venn diagram Von Neumann universe Von Neumann–Bernays–Gödel set theory W. Hugh Woodin Wacław Sierpiński Weak continuum hypothesis Well-formed formula Wetzel's problem Willard Van Orman Quine Zermelo set theory Zermelo–Fraenkel set theory Ω-consistency Ω-logic

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