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Proposition in mathematical logic
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:
Continuum_hypothesis
The term weak continuum hypothesis can be used to refer to the hypothesis that 2 ℵ 0 < 2 ℵ 1 {\displaystyle 2^{\aleph _{0}}<2^{\aleph _{1}}} , which is
Weak_continuum_hypothesis
The second continuum hypothesis, also called Luzin's hypothesis or Luzin's second continuum hypothesis, is the hypothesis that 2 ℵ 0 = 2 ℵ 1 {\displaystyle
Second_continuum_hypothesis
Cardinality of the set of real numbers
second smallest is ℵ 1 {\displaystyle \aleph _{1}} (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly
Cardinality_of_the_continuum
American mathematician (1934–2007)
was an American mathematician, best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set
Paul_Cohen
2005 studio album by Epoch of Unlight
The Continuum Hypothesis is the third full-length studio album by the American melodic death metal band Epoch of Unlight, released on March 8, 2005, through
The Continuum Hypothesis (album)
The_Continuum_Hypothesis_(album)
Mathematician (1845–1918)
believed the continuum hypothesis to be true and tried in vain for many years to prove it. His inability to prove the continuum hypothesis caused him considerable
Georg_Cantor
Philosophical notion
the hypothesis is neither true, nor false. It is then wrong to stipulate, a priori and for philosophical reasons, that the continuum hypothesis is true
Antiphilosophy
Infinite cardinal number
in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis (CH) is equivalent to the identity 2 ℵ 0 = ℵ 1 {\displaystyle 2^{\aleph
Aleph_number
Size of a possibly infinite set
Zermelo–Fraenkel set theory, such as the axiom of choice and the continuum hypothesis. For example, all infinite cardinal numbers are aleph numbers if
Cardinal_number
Subfield of mathematics
universe of set theory in which the continuum hypothesis must hold. In 1963, Paul Cohen showed that the continuum hypothesis cannot be proven from the axioms
Mathematical_logic
Basic framework of mathematics
reasons and that would decide the continuum hypothesis. Many large cardinal axioms were studied, but the hypothesis always remained independent from them
Foundations_of_mathematics
Standard system of axiomatic set theory
of choice from the remaining Zermelo-Fraenkel axioms and of the continuum hypothesis from ZFC. The consistency of a theory such as ZFC cannot be proved
Zermelo–Fraenkel_set_theory
Mathematical logician and philosopher
numbers. Gödel also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted Zermelo–Fraenkel set theory, assuming
Kurt_Gödel
Size of a set in mathematics
cardinality ℵ 1 {\displaystyle \aleph _{1}} is known as the continuum hypothesis, which has been shown to be both unprovable and undisprovable in
Cardinality
Branch of physics
continuum hypothesis fails can be solved using statistical mechanics or rarefied gas dynamics. To determine whether or not the continuum hypothesis applies
Fluid_mechanics
Limitative results in mathematical logic
extra axiom stating that there are no endpoints in the order. The continuum hypothesis is a statement in the language of ZFC that is not provable within
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Collection of mathematical objects
set theory with the continuum hypothesis added as a further axiom, and the set theory with the negation of the continuum hypothesis added. Informally,
Set_(mathematics)
Extinct Celtic languages of Iberia
developed into -bl- in names like Ableca. The Western Hispano-Celtic continuum hypothesis received little support from linguists, who have widely rejected
Hispano-Celtic_languages
Branch of mathematics that studies sets
the continuum hypothesis or the axiom of choice, the inner model L constructed inside the original model will satisfy both the generalized continuum hypothesis
Set_theory
Problem in set theory
of the continuum hypothesis implies the Suslin hypothesis. The Suslin hypothesis is also independent of both the generalized continuum hypothesis (proved
Suslin's_problem
Axiom in set theory
{\displaystyle {\texttt {AX}}} is equivalent to the negation of the continuum hypothesis (CH). Sierpiński's theorem answered a question of Hugo Steinhaus
Freiling's_axiom_of_symmetry
set theoretic statements are independent of ZFC, among others: the continuum hypothesis or CH (Gödel produced a model of ZFC in which CH is true, showing
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Set theory concept
1 {\displaystyle {\mathfrak {c}}=\aleph _{1}} is the well-known continuum hypothesis, which was shown to be consistent with the standard ZFC axioms for
Cardinal characteristic of the continuum
Cardinal_characteristic_of_the_continuum
Field in mathematics similar to the real numbers
assume the generalized continuum hypothesis. If the continuum hypothesis holds, all real closed fields with cardinality of the continuum and having the η1
Real_closed_field
Infinite Cardinal number
{\displaystyle \aleph _{0},\aleph _{1},\dots } ), but unless the generalized continuum hypothesis is true, there are numbers indexed by ℵ {\displaystyle \aleph } that
Beth_number
23 mathematical problems stated in 1900
ISBN 978-0387946740. Cohen, Paul J. (15 December 1963). "The independence of the Continuum Hypothesis, [part I]". Proceedings of the National Academy of Sciences of the
Hilbert's_problems
Set theory concept
singular cardinals hypothesis (SCH) arose from the question of whether the least cardinal number for which the generalized continuum hypothesis (GCH) might fail
Singular_cardinals_hypothesis
Possible axiom for set theory in mathematics
axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslin's hypothesis, and the existence of an analytical (in fact,
Axiom_of_constructibility
Technique invented by Paul Cohen for proving consistency and independence results
in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. It has been considerably reworked
Forcing_(mathematics)
Axiom of set theory
statement that is independent of ZF. For example, the generalized continuum hypothesis (GCH) is not only independent of ZF, but also independent of ZFC
Axiom_of_choice
Yes-or-no question that cannot ever be solved by a computer
of undecidable statements (in the first sense of the term): The continuum hypothesis can neither be proved nor refuted in ZFC (the standard axiomatization
Undecidable_problem
Set theory concept
ISBN 0-486-66637-9. Cohen, Paul Joseph (2008) [1966]. Set theory and the continuum hypothesis. Mineola, New York: Dover Publications. ISBN 978-0-486-46921-8. Gödel
Von_Neumann_universe
Area of mathematical logic
axioms of Zermelo–Fraenkel set theory, and is true if the generalised continuum hypothesis holds. Ultraproducts are used as a general technique for constructing
Model_theory
Axiom in the mathematical field of set theory
theory. It is implied by the continuum hypothesis, but it is consistent with ZFC and the negation of the continuum hypothesis. Informally, it says that all
Martin's_axiom
The real numbers or their cardinality
natural numbers. The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis is sometimes stated by saying that no
Continuum_(set_theory)
technique, which was developed to prove the independence of the continuum hypothesis from ZFC. Showing that an axiom is independent is often helpful for
Axiom_independence
Statement that is taken to be true
Furthermore, using techniques of forcing (Cohen) one can show that the continuum hypothesis (Cantor) is independent of the Zermelo–Fraenkel axioms. Thus, even
Axiom
Ability to learn vocalization
learning continuum hypothesis by Erich Jarvis and Gustavo Arriaga. Based on the apparent variations seen in various studies, the continuum hypothesis reclassifies
Vocal_learning
German logician and mathematician (1871–1953)
coming century. The first of these, a problem of set theory, was the continuum hypothesis introduced by Cantor in 1878, and in the course of its statement
Ernst_Zermelo
Particular class of sets which can be described entirely in terms of simpler sets
paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this paper, he proved that the constructible universe is an
Constructible_universe
Proof in set theory
for the comprehension scheme. Cantor's first uncountability proof Continuum hypothesis Controversy over Cantor's theory Diagonal lemma the diagonalisation
Cantor's_diagonal_argument
Term in mathematical logic
that ZF is consistent: The axiom of choice The continuum hypothesis and the generalized continuum hypothesis The Suslin conjecture The following statements
Independence (mathematical logic)
Independence_(mathematical_logic)
Method of deriving conclusions
March 2025. Williamson, Jon; Russo, Federica (2010). Key Terms in Logic. Continuum. ISBN 978-1-84706-114-0. Zalta, Edward N. (2024). "Gottlob Frege". The
Rule_of_inference
Infinite set that is not countable
1 = ℶ 1 {\displaystyle \aleph _{1}=\beth _{1}} is now called the continuum hypothesis, and is known to be independent of the Zermelo–Fraenkel axioms for
Uncountable_set
Topics referred to by the same term
real line Continuum (topology), a nonempty compact connected metric space (sometimes Hausdorff space) Continuum hypothesis, the hypothesis that no infinite
Continuum
Set theory concept
doi:10.1016/0003-4843(78)90031-1. Woodin, W. Hugh (2001). "The continuum hypothesis, part II". Notices of the American Mathematical Society. 48 (7):
Large_cardinal
Type of infinite structure
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
O-minimal_theory
Question in abstract algebra
even if one assumes the continuum hypothesis. In fact, it remains undecidable even under the generalised continuum hypothesis. The Whitehead conjecture
Whitehead_problem
In mathematics, a statement that has been proven
conjecture). The term hypothesis is also used in this sense (e.g. Riemann hypothesis), which should not be confused with "hypothesis" as the premise of a
Theorem
Number representing a continuous quantity
strictly smaller than c {\displaystyle {\mathfrak {c}}} is known as the continuum hypothesis (CH). The axiom system most commonly used in mathematics, Zermelo-Fraenkel
Real_number
generalized continuum hypothesis below a strongly compact cardinal imply the generalized continuum hypothesis everywhere? Does the generalized continuum hypothesis
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Element of a nonstandard model of the reals, which can be infinite or infinitesimal
This question turns out to be equivalent to the continuum hypothesis; in ZFC with the continuum hypothesis we can prove this field is unique up to order
Hyperreal_number
Perspective of mathematical philosophy
multiverse views is the attitude to the continuum hypothesis. In the universe view the continuum hypothesis is a meaningful question that is either true
Multiverse_(set_theory)
complicated: the answer to Wetzel's question is yes if and only if the continuum hypothesis is false. That is, the existence of an uncountable set of functions
Wetzel's_problem
Form of logic that allows quantification over predicates
only model is the real numbers if the continuum hypothesis holds and that has no model if the continuum hypothesis does not hold. This theory consists of
Second-order_logic
Concept in topology
of N* (this does not need the continuum hypothesis, but is less interesting in its absence). If the continuum hypothesis holds then N* is the unique Parovicenko
Stone–Čech_compactification
Polish mathematician (1882–1969)
contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and topology. He published
Wacław_Sierpiński
Paradox in set theory
Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one
Russell's_paradox
Mathematics award
independence in set theory of the axiom of choice and of the generalized continuum hypothesis. The latter problem was the first of Hilbert's problems of the 1900
Fields_Medal
System of mathematical set theory
relative consistency proof of the axiom of choice and the generalized continuum hypothesis. Classes have several uses in NBG: They produce a finite axiomatization
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Number that is larger than all finite numbers
way, there are no cardinals between aleph-null and aleph-one. The continuum hypothesis is the proposition that there are no intermediate cardinal numbers
Transfinite_number
Set of the elements not in a given subset
Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one
Complement_(set_theory)
Mathematical concept
{\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}} . The continuum hypothesis states that there is no cardinal number between the cardinality of
Infinity
Number of arguments required by a function
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Arity
American mathematician (born 1955)
particular, the continuum hypothesis would be true in this universe. In 2008, Woodin gave the Gödel Lecture titled The Continuum Hypothesis, the Conjecture
W._Hugh_Woodin
Deductive system in set theory
The theory he developed involves a controversial argument that the continuum hypothesis is false. Woodin's Ω-conjecture asserts that if there is a proper
Ω-logic
Concept in mathematics
ZF by Shoenfield's absoluteness theorem. ZFC with the generalized continuum hypothesis is a Π2 1-conservative extension of ZFC. With model-theoretic means
Conservative_extension
Any one of the distinct objects that make up a set in set theory
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Element_of_a_set
Mathematical set containing no elements
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Empty_set
Topics referred to by the same term
computer science, a containment hierarchy of classes of formal grammars Continuum hypothesis, in set theory Hyperbolic cosine, in mathematics, a hyperbolic function
CH
Mathematical theorem in set theory
class of forcing conditions over a model satisfying the generalized continuum hypothesis. The first two conditions in the theorem are necessary. Condition
Easton's_theorem
Set of elements in any of some sets
Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one
Union_(set_theory)
Combinatorial principle
(1972) that holds in the constructible universe and that implies the continuum hypothesis. Jensen extracted the diamond principle from his proof that the axiom
Diamond_principle
Mathematical set that can be enumerated
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Countable_set
Theorem in axiomatic set theory
consistency). Under this hypothesis cardinal exponentiation is simplified, though not to the extent of the generalized continuum hypothesis (which implies the
Gimel_function
Type of infinite number in set theory
inaccessible cardinal is a weakly inaccessible cardinal. The generalized continuum hypothesis implies that all weakly inaccessible cardinals are strongly inaccessible
Inaccessible_cardinal
Set of all things that may be the input of a mathematical function
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Domain_of_a_function
Set of elements common to all of some sets
Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one
Intersection_(set_theory)
of the real line Continuum (topology), a nonempty compact connected metric space (sometimes a Hausdorff space) Continuum hypothesis, a conjecture of Georg
List of continuity-related mathematical topics
List_of_continuity-related_mathematical_topics
Mathematical set formed from two given sets
Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one
Cartesian_product
Set whose elements all belong to another set
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Subset
One-to-one correspondence
Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one
Bijection
Corollary 3.6, Lemma 3.7. Cohen, P. J. (1966). Set theory and the continuum hypothesis. Addison–Wesley. ISBN 978-0-8053-2327-6. Kunen, Kenneth (1980). Set
Standard_model_(set_theory)
continuum hypothesis implies that a Luzin space exists. Kunen (1977) showed that assuming Martin's axiom and the negation of the continuum hypothesis
Luzin_space
Branch of physics which studies the behavior of materials modeled as continuous media
called a continuum) rather than as discrete particles. Continuum mechanics deals with deformable bodies, as opposed to rigid bodies. A continuum model assumes
Continuum_mechanics
Mathematical concept for comparing objects
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Equivalence_relation
Aspect of psychosis
better understood on a spectrum. According to Craddock and Owen, the continuum extends from unipolar depression, bipolar affective, schizoaffective,
Kraepelinian_dichotomy
Impossible task in computing
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Entscheidungsproblem
Target set of a mathematical function
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Codomain
German mathematician
Mahlo introduced Mahlo cardinals in 1911. He also showed that the continuum hypothesis implies the existence of a Luzin set. Mahlo, Paul (1908), Topologische
Paul_Mahlo
Logical incompatibility between two or more propositions
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Contradiction
understood when considering specific examples, such as the "continuum hypothesis". The continuum hypothesis has been proven independent of the ZF axioms of set
List of philosophical problems
List_of_philosophical_problems
Undecidability of equality of real numbers
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Richardson's_theorem
British mathematician (born 1944)
1969 with a dissertation on Large Cardinals and the Generalized Continuum Hypothesis. Paris is known for his work on mathematical logic, in particular
Jeff_Paris_(mathematician)
Mathematical function such that every output has at least one input
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Surjective_function
Collection of sets in mathematics that can be defined based on a property of its members
Springer-Verlag Smullyan, Raymond M.; Fitting, Melvin (2010), Set Theory And The Continuum Problem, Dover Publications, ISBN 978-0-486-47484-7 Monk, Donald J. (1969)
Class_(set_theory)
Mathematical set of all subsets of a set
one-to-one correspondence with the set of real numbers (see Cardinality of the continuum). The power set of a set S, together with the operations of union, intersection
Power_set
Type of logical system
scholar", is a conditional statement with "x is a philosopher" as its hypothesis, and "x is a scholar" as its conclusion, which again needs specification
First-order_logic
Form of mathematical proof
The hypothesis in the induction step, that the statement holds for a particular n, is called the induction hypothesis or inductive hypothesis. To prove
Mathematical_induction
CONTINUUM HYPOTHESIS
CONTINUUM HYPOTHESIS
Girl/Female
Tamil
Continuous, Younger sister
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Continuous
Boy/Male
Tamil
Continuous
Boy/Male
Hindu
Continuous
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Continuous
Boy/Male
Gujarati, Hindu, Indian
Continuous
Girl/Female
Indian
Continuous, Younger sister
Boy/Male
Hindu, Indian
Continuer
Boy/Male
Tamil
Continuous
Girl/Female
Hindu, Indian, Marathi, Tamil, Telugu
Continuous Flow
Boy/Male
Arabic
Continual; Listing
Girl/Female
Tamil
Continuous, Younger sister
Girl/Female
Arabic, Muslim
Continues
Boy/Male
Hindu, Indian, Marathi
Continuous Extended
Girl/Female
Hindu, Indian
Continuous
Boy/Male
Tamil
Continuous
Girl/Female
Tamil
Prahasini | பà¯à®°à®¹à®¸à¯€à®¨à¯€Â
Continues smiling girl
Prahasini | பà¯à®°à®¹à®¸à¯€à®¨à¯€Â
Girl/Female
Latin
Perpetual; continual.
Boy/Male
Gujarati, Hindu, Indian, Marathi, Sanskrit
Continuous; Ongoing
Boy/Male
Hindu, Indian
Tone Continued
CONTINUUM HYPOTHESIS
CONTINUUM HYPOTHESIS
Boy/Male
Celtic
Lives near the yew tree estate.
Girl/Female
Tamil
Fair skinned
Boy/Male
British, English
Form of Reginald; Counsel Power
Girl/Female
Arabic, Bengali, Gujarati, Hindu, Indian, Kannada, Muslim, Punjabi, Sikh, Tamil
Smiling; Smile; Symbol of Happiness
Girl/Female
American, Australian, Christian, French, Latin
Precious Stone; A Gem; Plaything; Delight; Jewel
Boy/Male
British, English, French, German, Polish, Teutonic
Name of a Bishop; Prosperous; Wealth; Rich
Boy/Male
Australian, German, Teutonic
Hard-working Ruler; Home Ruler
Girl/Female
Tamil
The Moon
Surname or Lastname
English
English : from a pet form of the Old English personal name Goda, which was in part a byname and in part a short form of various compound names with the first element gÅd.
Boy/Male
Arabic, Australian
Handsome
CONTINUUM HYPOTHESIS
CONTINUUM HYPOTHESIS
CONTINUUM HYPOTHESIS
CONTINUUM HYPOTHESIS
CONTINUUM HYPOTHESIS
adv.
Constant; continual.
n.
One who, or that which, continues; esp., one who continues a series or a work; a continuer.
imp. & p. p.
of Continue
a.
Prolonged; continued.
a.
Uninterrupted; unbroken; continual; continued.
n.
One who continues; one who has the power of perseverance or persistence.
a.
Proceeding without interruption or cesstaion; continuous; unceasing; lasting; abiding.
p. p. & a.
Having extension of time, space, order of events, exertion of energy, etc.; extended; protracted; uninterrupted; also, resumed after interruption; extending through a succession of issues, session, etc.; as, a continued story.
a.
Without break, cessation, or interruption; without intervening space or time; uninterrupted; unbroken; continual; unceasing; constant; continued; protracted; extended; as, a continuous line of railroad; a continuous current of electricity.
v. t.
To retain; to suffer or cause to remain; as, the trustees were continued; also, to suffer to live.
n.
Basso continuo, or continued bass.
a.
Occuring in steady and rapid succession; very frequent; often repeated.
v. i.
To be steadfast or constant in any course; to persevere; to abide; to endure; to persist; to keep up or maintain a particular condition, course, or series of actions; as, the army continued to advance.
a.
Not deviating or varying from uninformity; not interrupted; not joined or articulated.
a.
Unceasing; continual.
v. t. & i.
To continue anew.
n.
Thread; continuous line.
n.
A continuous fever.
a.
Continual; incessant; unintermitted.
p. pr. & vb. n.
of Continue