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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
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
2^{\aleph _{0}}<2^{\aleph _{1}}} , which is the negation of the second continuum hypothesis. It is equivalent to a weak form of ◊ on ℵ 1 {\displaystyle \aleph
Weak_continuum_hypothesis
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
Cardinality of the set of real numbers
\aleph _{0}} (aleph-null). The second smallest is ℵ 1 {\displaystyle \aleph _{1}} (aleph-one). The continuum hypothesis, which asserts that there are no
Cardinality_of_the_continuum
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
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
Standard system of axiomatic set theory
the continuum hypothesis from ZFC. The consistency of a theory such as ZFC cannot be proved within the theory itself, as shown by Gödel's second incompleteness
Zermelo–Fraenkel_set_theory
learner. In addition, Krashen (1982)'s Affective Filter Hypothesis holds that the acquisition of a second language is halted if the learner has a high degree
Theories of second-language acquisition
Theories_of_second-language_acquisition
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
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
Hypothesis that reality could be a computer simulation
The simulation hypothesis proposes that what one experiences as the real world is actually a simulated reality, such as a computer simulation in which
Simulation_hypothesis
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
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
Question in abstract algebra
the negation of the continuum hypothesis, Whitehead's problem cannot be resolved in ZFC. J. H. C. Whitehead, motivated by the second Cousin problem, first
Whitehead_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
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
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
Mathematical model combining space and time
space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime
Spacetime
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
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
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
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
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
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
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
In mathematics, a generalization of the real line
mathematical field of order theory, a continuum or linear continuum is a generalization of the real line. Formally, a linear continuum is a linearly ordered set S
Linear_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
Mathematical-logic system based on functions
(n, n + 1) can be defined as Φ := λp.PAIR (SECOND p) (SUCC (SECOND p)) Ψ := λfp.PAIR (SECOND p) (f (SECOND p)) which allows us to give perhaps the most
Lambda_calculus
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
Proof by Alan Turing
Computable Numbers, with an Application to the Entscheidungsproblem". It was the second proof (after Church's theorem) of the negation of Hilbert's Entscheidungsproblem;
Turing's_proof
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
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
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
3-volume treatise on mathematics, 1910–1913
"number" in the infinite sense (i.e. the continuum) cannot be described by the new theory proposed in PM Second Edition. Wittgenstein in his Lectures on
Principia_Mathematica
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
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
Mathematical set formed from two given sets
real numbers, called its coordinates. Usually, such a pair's first and second components are called its x and y coordinates, respectively (see picture)
Cartesian_product
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)
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)
Mathematical theory of data types
"S". "term elimination" rules define the other functions like "first", "second", and "R". "computation" rules specify how computation is performed with
Type_theory
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
Mathematician (1845–1918)
ordinal arithmetic are reviewed. Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for laying out his theory
Georg_Cantor
Pattern of romantic/sexual attraction based on sex/gender
bisexual orientation. A person's sexual orientation can be anywhere on a continuum, from exclusive attraction to the opposite sex to exclusive attraction
Sexual_orientation
Consistency of the axioms of arithmetic
In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that arithmetic is consistent
Hilbert's_second_problem
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
Diagram that shows all possible logical relations between a collection of sets
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Venn_diagram
Branch of the Indo-European language family
particularly innovative dialect separated from the Balto-Slavic dialect continuum and became ancestral to the Proto-Slavic language, from which all Slavic
Balto-Slavic_languages
ISBN 978-1-4082-0460-3. VanPatten, Bill; Benati, Alessandro G. (2010). Key Terms in Second Language Acquisition. London: Continuum. ISBN 978-0-8264-9914-1. v t e
Skill-based theories of second-language acquisition
Skill-based_theories_of_second-language_acquisition
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
Logical principle
two contradictories remounts, as I have said, also to Plato, though the Second Alcibiades, the dialogue in which it is most clearly expressed, must be
Law_of_excluded_middle
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
System of mathematical set theory
B\subseteq B} is another way of expressing that B is transitive. The inductive hypothesis then informs us that ∀ a ∈ A ∃ b ( a ∈ b ∧ ⋃ b ⊆ b ) {\displaystyle \forall
Kripke–Platek_set_theory
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
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Mathematical_object
Measure of algorithmic complexity
reasoning Kolmogorov structure function Levenshtein distance Manifold hypothesis Solomonoff's theory of inductive inference Sample entropy Rayo's number
Kolmogorov_complexity
Axioms for the natural numbers
induction is sometimes stated in the following form that uses a stronger hypothesis, making use of the order relation "≤": For any predicate φ, if φ(0) is
Peano_axioms
Mathematical set containing no elements
{\displaystyle \varnothing } ". The first compares elements of sets, while the second compares the sets themselves. Jonathan Lowe argues that while the empty
Empty_set
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
Hypothesis that human replicas elicit revulsion
human being and the emotional response to the object. The uncanny valley hypothesis predicts that an entity appearing almost human will elicit uncanny or
Uncanny_valley
Language family
legitimate scholarly arguments for both the Insular Celtic hypothesis and the P-/Q-Celtic hypothesis. Proponents of each schema dispute the accuracy and usefulness
Celtic_languages
Process of learning a second language
on the critical period hypothesis and learning strategies. In addition to acquisition, SLA explores language loss, or second-language attrition, and
Second-language_acquisition
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
Problem in computer science
known as Hilbert's problems) at the Second International Congress of Mathematicians in Paris. "Of these, the second was that of proving the consistency
Halting_problem
Axiom of set theory proposed by Peter Aczel in 1988
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Aczel's_anti-foundation_axiom
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
Paradox in set theory
is that TT relies on a strong higher-order logic, while Zermelo employed second-order logic, and ZFC can also be given a first-order formulation. The first-order
Russell's_paradox
Every set is smaller than its power set
strictly larger than the cardinality of the integers; see Cardinality of the continuum for details. The theorem is named for Georg Cantor, who first stated and
Cantor's_theorem
Symbol representing a mathematical object
almost exclusively to the arguments and the values of functions. In the second half of the 19th century, it appeared that the foundation of infinitesimal
Variable_(mathematics)
Language skill phenomenon
130). This hypothesis is more differentiated and complex than the regression hypothesis because it considers aspects from first- and second-language acquisition
Second-language_attrition
Input to a mathematical function
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Argument_of_a_function
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
One-to-one correspondence
correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set
Bijection
Complexity class used to classify decision problems
about the solution, which is generated in a nondeterministic way, while the second phase consists of a deterministic algorithm that verifies whether the guess
NP_(complexity)
Function computable with bounded loops
{\displaystyle h} is fed the "current" value of the for-loop's index. The second parameter of h {\displaystyle h} is fed the result of the for-loop's previous
Primitive_recursive_function
Theorem in mathematical logic
{\displaystyle F} is a finite field or the algebraic closure of such a field. A second application of the compactness theorem shows that any theory that has arbitrarily
Compactness_theorem
Theorem for proving more complex theorems
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Lemma_(mathematics)
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
Hypothesis that humans are born with knowledge of linguistic structure
In linguistics, the innateness hypothesis, also known as the nativist hypothesis, holds that humans are born with at least some knowledge of linguistic
Innateness_hypothesis
resonances (this is the so-called flat continuum hypothesis). If one succeeds in translating the flat continuum hypothesis in a mathematical form, it is possible
Feshbach–Fano_partitioning
Informal set theories
second element, and having the fundamental property that, two ordered pairs are equal if and only if their first elements are equal and their second elements
Naive_set_theory
Astronomical theory about the Solar System
The nebular hypothesis is the most widely accepted model in the field of cosmogony to explain the formation and evolution of the Solar System (as well
Nebular_hypothesis
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
Additional mathematical object
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Mathematical_structure
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Abstract_model_theory
School of thought in philosophy of mathematics
x can be treated in his logic, Russell proposed, as a kind of working hypothesis, that all such impredicative definitions have predicative definitions
Logicism
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)
Hypothetical source of gospel contents
Two-Source Hypothesis: A Critical Appraisal. Mercer University Pr. p. 241. Allison Jr., Dale (1997). The Jesus Tradition in Q. Continuum International
Q_source
Process of repeating items in a self-similar way
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Recursion
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Institutional_model_theory
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)
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
Group of theories relating to early Christian history
the two-source hypothesis, scholarly interest in the Hebrew gospel hypothesis dwindled. Modern variants of the Hebrew gospel hypothesis survive, but have
Hebrew_Gospel_hypothesis
Relationship where one statement follows from another
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Logical_consequence
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Atomic model (mathematical logic)
Atomic_model_(mathematical_logic)
Mathematical concept
<\beta \rangle } , where β is an ordinal with the cardinality of the continuum. Let v0 equal r0. Then let v1 equal rα1, where α1 is least such that rα1 − v0
Transfinite_induction
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
Template that specifies one or more axioms
this second-order sentence, F {\displaystyle F} is a genuine variable ranging over properties or classes, not a metalinguistic placeholder. The second-order
Axiom_schema
Function that preserves distinctness
number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck
Injective_function
Impossible task in computing
cylindrical algebraic decomposition. Automated theorem proving Hilbert's second problem Oracle machine Turing's proof David Hilbert and Wilhelm Ackermann
Entscheidungsproblem
SECOND CONTINUUM-HYPOTHESIS
SECOND CONTINUUM-HYPOTHESIS
Boy/Male
Hindu
Continuous
Girl/Female
Latin
Perpetual; continual.
Female
English
From the name of the state of Arizona in the United States of America, a place considered sacred by the Native Americans. It was named after Sedona Miller Schnebly (1877-1950), the wife of the city's first postmaster. Meaning unknown.
Girl/Female
Hindu, Indian, Marathi, Tamil, Telugu
Continuous Flow
Girl/Female
Tamil
Second
Boy/Male
Tamil
Continuous
Girl/Female
Arabic, Muslim
Continues
Male
English
Variant spelling of Middle English Estmond, ESMOND means "gracious protector."Â
Boy/Male
Gujarati, Hindu, Indian
Continuous
Boy/Male
Indian
Second
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Continuous
Boy/Male
Tamil
Continuous
Female
English
Anglicized form of Scottish Gaelic Seònaid, SEONA means "God is gracious."
Boy/Male
Hindu, Indian
Continuer
Girl/Female
Biblical
Second.
Girl/Female
Indian
Second
Girl/Female
Hindu, Indian
Continuous
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Continuous
Boy/Male
Tamil
Continuous
Surname or Lastname
English
English : from Richward, a Norman personal name composed of the Germanic elements rīc ‘power(ful)’ + ward ‘guard’.French : from Old French record, recort ‘recollection’, ‘account’, ‘testimony’, and by extension ‘witness’, hence perhaps a nickname for someone who had given evidence in a court of law, or a metonymic occupational name for a clerk who recorded court proceedings.New England variant of French Ricard, reflecting an Americanized spelling of the Canadian pronunciation.
SECOND CONTINUUM-HYPOTHESIS
SECOND CONTINUUM-HYPOTHESIS
Boy/Male
Indian, Punjabi, Sikh
Protected Victory
Boy/Male
Hindu
A brahmin in the epics
Boy/Male
Arabic, Muslim
An Effect; Impression
Boy/Male
Tamil
The Sun
Boy/Male
American, Australian, British, Chinese, Christian, Danish, Dutch, English, French, German, Indian, Netherlands, Scandinavian, Swedish, Swiss, Teutonic
Ruler of the People; Form of Derek; First of the People; King of Nations
Boy/Male
Arabic, Muslim
Friends
Surname or Lastname
English
English : regional name for someone from Burgundy, Old French Bourgogne (see Burgoyne).Swiss German (Bürgin) : from a pet form of the personal name Burkhard (see Burkhart).
Girl/Female
Tamil
Girl/Female
Hindu, Indian, Sanskrit, Tamil
Goddess of Victory
Girl/Female
English
lime tree; linden tree; beautiful.
SECOND CONTINUUM-HYPOTHESIS
SECOND CONTINUUM-HYPOTHESIS
SECOND CONTINUUM-HYPOTHESIS
SECOND CONTINUUM-HYPOTHESIS
SECOND CONTINUUM-HYPOTHESIS
v. t.
To retain; to suffer or cause to remain; as, the trustees were continued; also, to suffer to live.
a.
Of the second size, rank, quality, or value; as, a second-rate ship; second-rate cloth; a second-rate champion.
n.
One who seconds or supports what another attempts, affirms, moves, or proposes; as, the seconder of an enterprise or of a motion.
a.
Uninterrupted; unbroken; continual; continued.
a.
The sixtieth part of a minute of time or of a minute of space, that is, the second regular subdivision of the degree; as, sound moves about 1,140 English feet in a second; five minutes and ten seconds north of this place.
a.
Having the power of second-sight.
a.
Proceeding without interruption or cesstaion; continuous; unceasing; lasting; abiding.
n.
The second part in a concerted piece.
a.
To follow or attend for the purpose of assisting; to support; to back; to act as the second of; to assist; to forward; to encourage.
a.
Of the rank or degree below the best highest; inferior; second-rate; as, a second-class house; a second-class passage.
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.
adv.
Secondly; in the second place.
a.
Being of the same kind as another that has preceded; another, like a protype; as, a second Cato; a second Troy; a second deluge.
adv.
In the second place.
n.
One who, or that which, continues; esp., one who continues a series or a work; a continuer.
n.
Basso continuo, or continued bass.
imp. & p. p.
of Continue
n.
One who continues; one who has the power of perseverance or persistence.
n.
The second part in a concerted piece; -- often popularly applied to the alto.
imp. & p. p.
of Second