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Every set is smaller than its power set
question marks, boxes, or other symbols. In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A {\displaystyle
Cantor's_theorem
Theorem in set theory
The theorem is named after Ernst Schröder and Felix Bernstein. It is also known as the Cantor–Bernstein theorem or Cantor–Schröder–Bernstein theorem, after
Schröder–Bernstein_theorem
Mathematician (1845–1918)
real numbers are more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He
Georg_Cantor
Proof in set theory
R. A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: for every set S, the power set of S—that is, the set of all
Cantor's_diagonal_argument
On decreasing nested sequences of non-empty compact sets
Cantor's intersection theorem, also called Cantor's nested intervals theorem, refers to two closely related theorems in general topology and real analysis
Cantor's_intersection_theorem
About mathematical infinity
philosophers. Cantor's theorem implies that there are sets having cardinality greater than the infinite cardinality of the set of natural numbers. Cantor's argument
Controversy over Cantor's theory
Controversy_over_Cantor's_theory
Mathematical theorem
Heine–Cantor theorem states that a continuous function between two metric spaces is uniformly continuous if its domain is compact. The theorem is named
Heine–Cantor_theorem
Theorem in category theory
mathematics and logic, such as Cantor's diagonal argument, Cantor's theorem, Russell's paradox, Gödel's first incompleteness theorem, Turing's solution to the
Lawvere's_fixed-point_theorem
Topics referred to by the same term
Look up Cantor's theorem in Wiktionary, the free dictionary. Cantor's theorem is a fundamental result in mathematical set theory. Cantor's theorem may also
Cantor's theorem (disambiguation)
Cantor's_theorem_(disambiguation)
Mathematical set containing all objects
of all sets, provided that both exist. However, this conflicts with Cantor's theorem that the power set of any set (whether infinite or not) always has
Universal_set
Paradox in set theory
In set theory, Cantor's paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal
Cantor's_paradox
Uniqueness of countable dense linear orders
mathematics, Cantor's isomorphism theorem states that every two nonempty countable dense unbounded linear orders are order-isomorphic. The theorem is named
Cantor's_isomorphism_theorem
First article on transfinite set theory
Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One
Cantor's first set theory article
Cantor's_first_set_theory_article
Mathematical set of all subsets of a set
the power set must be larger than the original set). In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably
Power_set
Size of a set in mathematics
numbers are proven to be uncountable by so-called diagonal arguments. Cantor's theorem generalizes these arguments to show there is an infinite hierarchy
Cardinality
Limitative results in mathematical logic
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
(mathematical logic) Cantor–Bernstein–Schröder theorem (set theory, cardinal numbers) Cantor's theorem (set theory) Church–Rosser theorem (lambda calculus)
List_of_theorems
Impossible task in computing
impossible by Alonzo Church and Alan Turing in 1936. By the completeness theorem of first-order logic, a statement is universally valid if and only if it
Entscheidungsproblem
Equivalence between synthetic and analytic geometries
algorithm to solve any first-order problem in Euclidean geometry. Cantor's theorem Artin, Emil (1988) [1957], Geometric Algebra, Wiley Classics Library
Cantor–Dedekind_axiom
Paradox in set theory
Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction (to Cantor's theorem), as
Russell's_paradox
Branch of mathematics that studies sets
This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. Cantor introduced
Set_theory
Proposition in mathematical logic
under the then-undeveloped axiom of choice. Cantor initially presented the weak continuum hypothesis as a theorem, but did not give a proof and later became
Continuum_hypothesis
In mathematics, a statement that has been proven
mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses
Theorem
Existence and cardinality of models of logical theories
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf
Löwenheim–Skolem_theorem
Set of points on a line segment with certain topological properties
{2}}} . By a theorem of L. E. J. Brouwer, this is equivalent to being perfect, nonempty, compact, metrizable and zero-dimensional. The Cantor ternary set
Cantor_set
Theorem in set theory
{\displaystyle \kappa } . Thus, Kőnig's theorem gives us a proof of Cantor's theorem. (Historically of course Cantor's theorem was proved much earlier.) One way
Kőnig's_theorem_(set_theory)
Topics referred to by the same term
employed in proofs. The following theorems are notable examples: Cantor's diagonal argument (the earliest) Cantor's theorem Russell's paradox Curry's paradox
Diagonal_argument
Theorem for proving more complex theorems
also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however
Lemma_(mathematics)
Burnside's lemma Cantor's theorem Cantor–Bernstein–Schroeder theorem Cayley's formula Cayley's theorem Clique problem (to do) Compactness theorem (very compact
List_of_mathematical_proofs
System of mathematical set theory
class. Zermelo's paper may be the first to mention the name "Cantor's theorem". Cantor's theorem: "If M is an arbitrary set, then always M < P(M) [the power
Zermelo_set_theory
Theorem in mathematical logic
compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important
Compactness_theorem
Modern application of infinitesimals
a compact interval I is necessarily uniformly continuous (the Heine–Cantor theorem) admits a succinct hyperreal proof. Let x, y be hyperreals in the natural
Nonstandard_calculus
Mathematical set that can be enumerated
{P}}(A)} . A proof is given in the article Cantor's theorem. As an immediate consequence of this and the Basic Theorem above we have: Proposition—The set P
Countable_set
Fundamental theorem in mathematical logic
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability
Gödel's_completeness_theorem
Subfield of automated reasoning and mathematical logic
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving
Automated_theorem_proving
Logical connective AND
incompleteness theorems Tarski's undefinability Banach–Tarski paradox Cantor's theorem – paradox – diagonal argument Compactness Halting problem Lindström's
Logical_conjunction
Axiom of set theory
391–392. Banaschewski, Bernhard; Moore, Gregory H. (1990). "The dual Cantor-Bernstein theorem and the partition principle". Notre Dame Journal of Formal Logic
Axiom_of_choice
Problem in computer science
Minsky notes: ...the magnitudes involved should lead one to suspect that theorems and arguments based chiefly on the mere finiteness [of] the state diagram
Halting_problem
Mathematical logic concept
of the Löwenheim–Skolem theorem; Thoralf Skolem was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity
Skolem's_paradox
Theorem that arithmetical truth cannot be defined in arithmetic
Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Yes-or-no question that cannot ever be solved by a computer
are quite similar. In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the halting problem. This
Undecidable_problem
One-to-one correspondence
its inverse is the positive square root function. By Schröder–Bernstein theorem, given any two sets X and Y, and two injective functions f: X → Y and g:
Bijection
Informal set theories
they do exclude some paradoxes, like Russell's paradox. Based on Gödel's theorem, it is just not known – and never can be – if there are no paradoxes at
Naive_set_theory
Set whose elements all belong to another set
incompleteness theorems Tarski's undefinability Banach–Tarski paradox Cantor's theorem – paradox – diagonal argument Compactness Halting problem Lindström's
Subset
On the dimension of vector space duals
The Erdős–Kaplansky theorem is a theorem from linear algebra. The theorem makes a fundamental statement about the dimension of the dual spaces of infinite-dimensional
Erdős–Kaplansky_theorem
Undecidability of equality of real numbers
In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, π, ln 2
Richardson's_theorem
Process of repeating items in a self-similar way
this is a theorem guaranteeing that recursively defined functions exist. Given a set X, an element a of X and a function f: X → X, the theorem states that
Recursion
Form of logic that allows quantification over predicates
not the set of all subsets of the set of all internal numbers (since Cantor's theorem implies that the set of all subsets of a countably infinite set is
Second-order_logic
Property in descriptive set theory
uncountable set of reals has the cardinality of the continuum. The Cantor–Bendixson theorem states that closed sets of a Polish space X have the perfect set
Perfect_set_property
Method of deriving conclusions
inferential steps and often use various rules of inference to establish the theorem they intend to demonstrate. Rules of inference are definitory rules—rules
Rule_of_inference
Term in set theory
proofwiki.org. Retrieved 2019-11-16. "Theorem 36: the Cantor set is an uncountable set with zero measure". Theorem of the week. 2010-09-30. Retrieved 2019-11-16
Almost
Mathematical set containing no elements
example, Cantor defined two sets as being disjoint if their intersection has an absence of points; however, it is debatable whether Cantor viewed O {\displaystyle
Empty_set
Standard system of axiomatic set theory
shown by Gödel's second incompleteness theorem. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. However
Zermelo–Fraenkel_set_theory
3-volume treatise on mathematics, 1910–1913
set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems from real analysis were not included, but by the end of the third volume
Principia_Mathematica
function Cantor set Cantor space Cantor tree surface Cantor's back-and-forth method Cantor's diagonal argument Cantor's intersection theorem Cantor's isomorphism
List of things named after Georg Cantor
List_of_things_named_after_Georg_Cantor
Establishment of a theorem using inference from the axioms
the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is generally effective, but there may be no method
Formal_proof
Subfield of mathematics
argument, and used this method to prove Cantor's theorem that no set can have the same cardinality as its powerset. Cantor believed that every set could be well-ordered
Mathematical_logic
Axiomatic set theory devised by W.V.O. Quine
universal set, so it must be that Cantor's theorem (in its original form) does not hold in NF. Indeed, the proof of Cantor's theorem uses the diagonalization argument
New_Foundations
Set of elements in any of some sets
Science & Business Media. ISBN 9781475716450. "MathCS.org - Real Analysis: Theorem 1.1.4: De Morgan's Laws". mathcs.org. Archived from the original on 2024-11-10
Union_(set_theory)
Cardinality of the set of real numbers
Cantor's diagonal argument. A variation of Cantor's diagonal argument can be used to prove Cantor's theorem, which states that the cardinality of any set
Cardinality_of_the_continuum
Infinite cardinal number
theorem of ZFC. That it is independent of ZFC was demonstrated by Paul Cohen in 1963, when he showed conversely that the CH itself is not a theorem of
Aleph_number
Size of a possibly infinite set
happen with proper subsets of finite sets. However, a fundamental theorem due to Georg Cantor shows that it is possible for two infinite sets to have different
Cardinal_number
Mathematical concept
extension of mathematical induction to ordinal numbers. Its correctness is a theorem of ZF, and relies on the fact that the ordinal numbers are well-ordered
Transfinite_induction
Collection of mathematical objects
mathematical results are restated in terms of sets. For example, Euclid's theorem is often stated as "the set of the prime numbers is infinite". This wide
Set_(mathematics)
Set that is not a finite set
emphasizes how mathematicians such as Zermelo, Dedekind, Galileo, Kronecker, Cantor, and Bolzano investigated and influenced infinite set theory. Many of these
Infinite_set
Term in logic and deductive reasoning
Using the narrow definition of theorem, for sentences provable from no premises, weak soundness says that all theorems are tautologies. Strong soundness
Soundness
Paradox in set theory
1897, published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Georg Cantor. Bertrand Russell subsequently noticed
Burali-Forti_paradox
Uniform restraint of the change in functions
to the integers endowed with the usual Euclidean metric. The Heine–Cantor theorem asserts that every continuous function on a compact set is uniformly
Uniform_continuity
set P(S). Georg Cantor proved that the power set is always larger than the set, i.e., |P(S)| > |S|. A special case of Cantor's theorem is that the set
Paradoxes_of_set_theory
Study of computable functions and Turing degrees
and thus only countably many computable sets, but according to the Cantor's theorem, there are uncountably many sets of natural numbers. Although the halting
Computability_theory
Statement that is taken to be true
knowledge. They are accepted without demonstration. All other assertions (theorems, in the case of mathematics) must be proven with the aid of these basic
Axiom
Proof by Alan Turing
to the Entscheidungsproblem". It was the second proof (after Church's theorem) of the negation of Hilbert's Entscheidungsproblem; that is, the conjecture
Turing's_proof
Mathematical concept for comparing objects
the following three connected theorems hold: ~ partitions A into equivalence classes. (This is the Fundamental Theorem of Equivalence Relations, mentioned
Equivalence_relation
Mathematical proof expressed visually
third square. This process can be continued indefinitely. The Pythagorean theorem that a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} can be proven without
Proof_without_words
Complexity class used to classify decision problems
only known strict inclusions come from the time hierarchy theorem and the space hierarchy theorem, and respectively they are N P ⊊ N E X P T I M E {\displaystyle
NP_(complexity)
German logician and mathematician (1871–1953)
hypothesis introduced by Cantor in 1878, and in the course of its statement Hilbert also mentioned the need to prove the well-ordering theorem. Zermelo began to
Ernst_Zermelo
paradox Cantor's back-and-forth method Cantor's diagonal argument Cantor's first uncountability proof Cantor's paradox Cantor's theorem Cantor–Bernstein–Schroeder
List_of_set_theory_topics
Syntactically correct logical formula
mathematical software such as model checkers, automated theorem provers, interactive theorem provers) tend to retain of the notion of formula only the
Well-formed_formula
Set theory concept
incompleteness theorem. The observation that large cardinal axioms are linearly ordered by consistency strength is just that: an observation, not a theorem. (Without
Large_cardinal
Infinite set that is not countable
{\displaystyle \beth _{1}} (beth-one). The Cantor set is an uncountable subset of R {\displaystyle \mathbb {R} } . The Cantor set is a fractal and has Hausdorff
Uncountable_set
Template that specifies one or more axioms
sets, but unlike ZFC it can be finitely axiomatized: its class-existence theorem is obtained from finitely many class-existence axioms rather than from
Axiom_schema
Set with exactly one element
Uncountable Universal Theories Alternative Axiomatic Constructive Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel
Singleton_(mathematics)
Mathematical set formed from two given sets
Uncountable Universal Theories Alternative Axiomatic Constructive Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel
Cartesian_product
Any one of the distinct objects that make up a set in set theory
incompleteness theorems Tarski's undefinability Banach–Tarski paradox Cantor's theorem – paradox – diagonal argument Compactness Halting problem Lindström's
Element_of_a_set
Set of the elements not in a given subset
Uncountable Universal Theories Alternative Axiomatic Constructive Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel
Complement_(set_theory)
Collection of sets in mathematics that can be defined based on a property of its members
incompleteness theorems Tarski's undefinability Banach–Tarski paradox Cantor's theorem – paradox – diagonal argument Compactness Halting problem Lindström's
Class_(set_theory)
Whether a decision problem has an effective method to derive the answer
are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined. Zeroth-order logic (propositional logic)
Decidability_(logic)
There are equally many countable order types and real numbers
In set theory and order theory, the Cantor–Bernstein theorem states that the cardinality of the second type class, the class of countable order types,
Cantor–Bernstein_theorem
Pair of logical equivalences
logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference
De_Morgan's_laws
System of mathematical set theory
standard model of KP, then the set of ordinals in M is an admissible ordinal. Theorem: If A and B are sets, then there is a set A×B that consists of all ordered
Kripke–Platek_set_theory
Set theory concept
ZF sets is equal to the cumulative hierarchy"—not a definition, but a theorem equivalent to the axiom of regularity. Roitman states (without references)
Von_Neumann_universe
Model of (first-order) Peano arithmetic that contains non-standard numbers
the only countable dense linear order without endpoints (see Cantor's isomorphism theorem). So, the order type of the countable non-standard models is
Non-standard model of arithmetic
Non-standard_model_of_arithmetic
Pathological embedding of the sphere in 3D space
space that is "knotted" at every point. Cantor tree surface Wild knot Jordan curve theorem – The foundational theorem that the horned sphere generalizes (and
Alexander_horned_sphere
Axiom of Zermelo-Fraenkel set theory
the consistency of ZFC − Infinity and use Gödel's second incompleteness theorem.) The negation of the axiom of infinity cannot be derived from the rest
Axiom_of_infinity
Characteristic of some logical systems
having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is
Completeness_(logic)
Type of logical system
to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem. First-order logic is the standard for the formalization
First-order_logic
Set with algorithmic membership test
than a given natural number is computable. c.f. Gödel's incompleteness theorems; "On formally undecidable propositions of Principia Mathematica and related
Computable_set
Well-quasi-ordering of finite trees
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under
Kruskal's_tree_theorem
English philosopher and logician (1872–1970)
BBC Home Service in December 1948. John Newsome Crossley. A Note on Cantor's Theorem and Russell's Paradox, Australian Journal of Philosophy 51, 1973, 70–71
Bertrand_Russell
Logical incompatibility between two or more propositions
various rules that treat contradiction by considering theorems of classical logic that are not theorems of minimal logic. Each of these extensions leads to
Contradiction
CANTORS THEOREM
CANTORS THEOREM
Surname or Lastname
English
English : from an agent derivative of Anglo-Norman French cant ‘song’, applied as an occupational name for a singer in a chantry or a nickname for someone who had a good voice or who sang a lot.Americanized spelling of Kanter or Kantor.
Boy/Male
American, Australian, Chinese, French, German, Greek, Latin, Portuguese, Spanish
A Saint; Holy; The New House; Form of Santo
Boy/Male
Latin
Worthy of praise; of value. Saint Anthony is the patron sain of poor people. Famous Bearer:...
Boy/Male
Arthurian Legend
Foster father of Arthur.
Boy/Male
Spanish American Latin
Saint.
Girl/Female
Arabic
Small Bridge
Boy/Male
Latin
Worthy of praise; of value. Saint Anthony is the patron sain of poor people. Famous Bearer:...
Surname or Lastname
English
English : variant of Cater.
Girl/Female
Arabic Muslim
Bridge.
Boy/Male
Latin
Singer.
Boy/Male
British, English, Greek
Heart
Girl/Female
Native American
Spirit.
Boy/Male
Greek Latin
Beaver. Brother of Helen.
Surname or Lastname
English
English : variant spelling of Canter.German and Jewish (Ashkenazic) : variant spelling of Kantor.French (Picardy) : learned form of chantre ‘singer’. Compare Canter 1.
Girl/Female
Arabic, Muslim
Small Bridge
Girl/Female
Muslim
Small bridge
Boy/Male
Danish, French, German, Greek, Latin, Swedish
Brother of Helen; Braver
Male
Spanish
Portuguese and Spanish name SANTOS means "saints."Â This name is sometimes bestowed on a child to invoke the protection of the saints. It is also given to baby boys born on the Feast of All Saints.
Surname or Lastname
English
English : habitational name from places called Caistor, in Lincolnshire and Norfolk, Caister in Norfolk, or Castor in Cambridgeshire, all named with Old English cæster ‘Roman fort or town’.
Male
Celtic
, chief or king of a district or division.
CANTORS THEOREM
CANTORS THEOREM
Girl/Female
Indian
Perception, Intelligence, Life, Vigour
Boy/Male
Muslim/Islamic
The black cloth of the kaaba
Boy/Male
Hindu, Indian
Mace of Iron; Silver or Gold
Girl/Female
Indian, Punjabi, Sikh
Dear One
Girl/Female
Hindu, Indian, Marathi
Consent
Girl/Female
Celtic American Gaelic Irish
Famous.
Boy/Male
American, Australian, British, Christian, English, German
Walled; Stream Town; From the Welshman's Farm; From the Walled Town; Variant of Walter Rules; Spring Settlement
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu
King Among Actors; Name of Lord Shiva
Boy/Male
Tamil
Varaprada | வரபà¯à®°à®¤à®¾
Granter of wishes and boons
Surname or Lastname
English
English : nickname from the bird (Old English hrÅc), most likely given to a person with very dark hair or a dark complexion or to someone with a raucous voice.English : some early examples, such as Robert of ye Rook (London 1318) and Henry del Rook (Staffordshire 1332), point clearly to a local name of some kind. The first of these could be from a house sign, the second may be a variant of Rock 1.German : from a short form of a Germanic personal name formed with hrok, of uncertain origin; perhaps a cognate of 1 or from Middle High German rÅhen ‘to cry or yell (in battle)’ or Old High German ruoh ‘intent’.Perhaps an altered spelling of German Ruck.
CANTORS THEOREM
CANTORS THEOREM
CANTORS THEOREM
CANTORS THEOREM
CANTORS THEOREM
n.
A song or canto
p. pr. & vb. n.
of Canton
v. t.
To cause, as a horse, to go at a canter; to ride (a horse) at a canter.
a.
Of or pertaining to a canton or cantons; of the nature of a canton.
imp. & p. p.
of Canton
v. i.
The canto, cantus, or soprano voice; the treble.
p. pr. & vb. n.
of Canter
v. i.
To move in a canter.
n.
One who cants or whines; a beggar.
a.
Between centers.
n.
A civil officer in some Swiss cantons.
pl.
of Canto
v. i.
To divide into cantons or small districts.
n.
A chief magistrate in some of the Swiss cantons.
imp. & p. p.
of Canter
a.
Of or pertaining to a cantor; as, the cantoris side of a choir; a cantoris stall.
a.
Of or belonging to a cantor.
pl.
of Canthus
n.
One who caters.