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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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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)
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
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)
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
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)
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
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
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
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)
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
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
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
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
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
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)
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
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)
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
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
used forcing to establish that ZFC, if consistent, cannot prove the continuum hypothesis, which states that there are exactly ℵ 1 {\displaystyle \aleph _{1}}
Generic_filter
Mathematical operation with two operands
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Binary_operation
Mathematical use of "there exists"
\,Q(x))} A rule of inference is a rule justifying a logical step from hypothesis to conclusion. There are several rules of inference which utilize the
Existential_quantification
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
{\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}} . The continuum hypothesis states that there is no cardinal number between the cardinality of
Infinity
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
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
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
Type of cardinal number in mathematics
of the continuum, whose value in ZFC may be any uncountable cardinal of uncountable cofinality (see Easton's theorem). The continuum hypothesis postulates
Regular_cardinal
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)
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
Problem in computer science
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Halting_problem
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
2005 studio album by Epoch of Unlight
The Continuum Hypothesis is the 3rd full-length studio album released by the Melodic death/Black metal band Epoch of Unlight. It is the first to feature
The Continuum Hypothesis (album)
The_Continuum_Hypothesis_(album)
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
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 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
Symbol representing a property or relation in logic
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Predicate_(logic)
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
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
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
Evolutionary theory
postulated by the stress gradient hypothesis and the mutualism-parasitism continuum hypothesis. An example of the Red King hypothesis is the microbialite and coral
Red_King_hypothesis
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
Function, homomorphism, or morphism
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Map_(mathematics)
uniformization Axiom of real determinacy Von Neumann–Bernays–Gödel axioms Continuum hypothesis and its generalization Freiling's axiom of symmetry Axiom of determinacy
List_of_axioms
theory containing the arithmetic of infinite cardinal numbers and the continuum hypothesis. 1895 – Henri Poincaré publishes the paper "Analysis Situs," which
Timeline_of_mathematics
Logic that allows infinitely long proofs
infinitary logic named Ω-logic is complete promises to throw light on the continuum hypothesis. As a language with infinitely long formulae is being presented,
Infinitary_logic
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
Idea advanced by Ufologists
The interdimensional UFO hypothesis (IUH) is the proposal that unidentified flying object (UFO) sightings are the result of experiencing other "dimensions"
Interdimensional UFO hypothesis
Interdimensional_UFO_hypothesis
Concept in mathematics
theorem. Z F C {\displaystyle {\mathsf {ZFC}}} with the generalized continuum hypothesis is a Π 1 2 {\displaystyle \Pi _{1}^{2}} -conservative extension of
Conservative_extension
Thesis on the nature of computability
hypothesis—a point emphasized by Post and by Church. If we consider the thesis and its converse as definition, then the hypothesis is an hypothesis about
Church–Turing_thesis
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
Neurological disorder
recovery is possible along the semantic pathway. Friedman justifies the continuum hypothesis with two sets of evidence. The first involves five patients who started
Deep_dyslexia
Algebraic manipulation of "true" and "false"
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Boolean_algebra
Mathematical set containing all objects
Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one
Universal_set
Theorem in set theory
Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one
Schröder–Bernstein_theorem
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
Logic theorem
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Law_of_noncontradiction
CONTINUUM HYPOTHESIS
CONTINUUM HYPOTHESIS
Boy/Male
Arabic
Continual; Listing
Girl/Female
Arabic, Muslim
Continues
Boy/Male
Gujarati, Hindu, Indian, Marathi, Sanskrit
Continuous; Ongoing
Girl/Female
Tamil
Continuous, Younger sister
Girl/Female
Indian
Continuous, Younger sister
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Continuous
Boy/Male
Tamil
Continuous
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Continuous
Boy/Male
Hindu, Indian
Tone Continued
Boy/Male
Hindu, Indian, Marathi
Continuous Extended
Girl/Female
Tamil
Continuous, Younger sister
Boy/Male
Tamil
Continuous
Girl/Female
Latin
Perpetual; continual.
Boy/Male
Hindu, Indian
Continuer
Girl/Female
Tamil
Prahasini | பà¯à®°à®¹à®¸à¯€à®¨à¯€Â
Continues smiling girl
Prahasini | பà¯à®°à®¹à®¸à¯€à®¨à¯€Â
Boy/Male
Hindu
Continuous
Boy/Male
Gujarati, Hindu, Indian
Continuous
Girl/Female
Hindu, Indian
Continuous
Boy/Male
Tamil
Continuous
Girl/Female
Hindu, Indian, Marathi, Tamil, Telugu
Continuous Flow
CONTINUUM HYPOTHESIS
CONTINUUM HYPOTHESIS
Boy/Male
Hindu, Indian
Long Cherished Desire; Idea; Resolution
Girl/Female
Arabic, Muslim
Successful; Victorious; Triumph
Girl/Female
Arabic
Benefit.
Surname or Lastname
English
English : variant of Darby.
Boy/Male
Indian
The light
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu
Pleasure
Boy/Male
Hindu
Female
Russian
(Катенька) Diminutive form of Russian Ekaterina and Yekaterina, KATENKA means "pure."
Girl/Female
Indian, Punjabi, Sikh
Victorious Army of God in Heaven
Boy/Male
Hindu, Indian, Marathi
An Efficient Architect
CONTINUUM HYPOTHESIS
CONTINUUM HYPOTHESIS
CONTINUUM HYPOTHESIS
CONTINUUM HYPOTHESIS
CONTINUUM HYPOTHESIS
a.
Uninterrupted; unbroken; continual; continued.
imp. & p. p.
of Continue
n.
Basso continuo, or continued bass.
a.
Proceeding without interruption or cesstaion; continuous; unceasing; lasting; abiding.
a.
Not deviating or varying from uninformity; not interrupted; not joined or articulated.
a.
Occuring in steady and rapid succession; very frequent; often repeated.
a.
Unceasing; continual.
n.
One who, or that which, continues; esp., one who continues a series or a work; a continuer.
a.
Continual; incessant; unintermitted.
adv.
Constant; continual.
n.
Thread; continuous line.
n.
One who continues; one who has the power of perseverance or persistence.
p. pr. & vb. n.
of Continue
v. t. & i.
To continue anew.
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.
Prolonged; continued.
n.
A continuous fever.
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.
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.