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Second Continuum Hypothesis
The second continuum hypothesis, also called Luzin's hypothesis or Luzin's second continuum hypothesis, is the hypothesis that 2^=2^. It is the negation of a weakened form, 2^<2^, of the (CH). It was discussed by in 1935, although he did not claim to be the first to postulate it. The statement may also be called Luzin's hypothesis. The second continuum hypothesis is independent of with the |
Weak Continuum Hypothesis
The term weak continuum hypothesis can be used to refer to the hypothesis that 2^<2^, which is the negation of the second continuum hypothesis. It is equivalent to a weak form of ◊ on . F. Burton Jones proved that if it is true, then every separable normal Moore space is |
Continuum Hypothesis
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: 2^=\aleph_1, or even shorter with beth numbers: \beth_1 = \aleph_1. The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940. The name of the hypothesis comes from the term '' continuum'' for the real numbers. History Cantor believed the continuum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nikolai Luzin
Nikolai Nikolayevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlajɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 February 1950) was a Soviet and Russian mathematician known for his work in descriptive set theory and aspects of mathematical analysis with strong connections to point-set topology. He was the eponym of Luzitania, a loose group of young Moscow mathematicians of the first half of the 1920s. They adopted his set-theoretic orientation, and went on to apply it in other areas of mathematics. Life He started studying mathematics in 1901 at Moscow State University, where his advisor was Dmitri Egorov. He graduated in 1905. Luzin underwent great personal turmoil in the years 1905 and 1906, when his materialistic worldview had collapsed and he found himself close to suicide. In 1906 he wrote to Pavel Florensky, a former fellow mathematics student who was now studying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zermelo–Fraenkel Set Theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family (S_i)_ of nonempty sets (S_i as a nonempty set indexed with i), there exists an indexed set (x_i)_ such that x_i \in S_i for every i \in I. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. The axiom of choice is equivalent to the statement that every partition has a transversal. In many cases, a set created by choosing elements can be made without invoking the axiom of choice, particularly if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available — some distinguishing property that happens to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Cohen
Paul Joseph Cohen (April 2, 1934 – March 23, 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 theory, for which he was awarded a Fields Medal. Early life and education Cohen was born in Long Branch, New Jersey in 1934, into a Jewish family that had immigrated to the United States from what is now Poland; he grew up in Brooklyn.. He graduated in 1950, at age 16, from Stuyvesant High School in New York City. Cohen next studied at the Brooklyn College from 1950 to 1953, but he left without earning his bachelor's degree when he learned that he could start his graduate studies at the University of Chicago with just two years of college. At Chicago, Cohen completed his master's degree in mathematics in 1954 and his Doctor of Philosophy degree in 1958, under supervision of Antoni Zygmund. The title of his doctoral thesis was ''Topics in the Theory of Uniqueness of Trigo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin's Axiom
In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set 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 cardinals less than the cardinality of the continuum, 𝔠, behave roughly like ℵ0. The intuition behind this can be understood by studying the proof of the Rasiowa–Sikorski lemma. It is a principle that is used to control certain forcing arguments. Statement For a cardinal number ''κ'', define the following statement: ;MA(''κ''): For any partial order ''P'' satisfying the countable chain condition (hereafter ccc) and any set ''D'' = ''i''∈''I'' of dense subsets of ''P'' such that '', D, '' ≤ ''κ'', there is a filter ''F'' on ''P'' such that ''F'' ∩ ''D''''i'' is non- empty for every ''D''''i'' ∈ ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gregory H
Gregory may refer to: People and fictional characters * Gregory (given name), including a list of people and fictional characters with the given name * Gregory (surname), a surname *Gregory (The Walking Dead), fictional character from the walking dead * Gregory (Five Nights at Freddy's), main protagonist of '' Five Nights at Freddy's: Security Breach'' ** Places Australia *Gregory, a town in the Northern Territory *Gregory, Queensland, a town in the Shire of Burke **Electoral district of Gregory, Queensland, Australia * Gregory, Western Australia. United States *Gregory, South Dakota * Gregory, Tennessee * Gregory, Texas Outer space * Gregory (lunar crater) * Gregory (Venusian crater) Other uses * "Gregory" (''The Americans''), the third episode of the first season of the television series ''The Americans'' See also * Greg (other) * Greggory * Gregoire (other) * Gregor (other) * Gregores (other) * Gregorian (other) * Gregor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Juris Steprāns
The Juris (also ''Juri'', ''Yuri'') were a tribe of South American Indigenous people, formerly occupying the country between the rivers Içá (lower Putumayo) and Yapura, north-western Brazil. In ancient days they were the most powerful tribe of the district, but in 1820 their numbers did not exceed 2000. Owing to inter-marrying, the Juris are believed to have been extinct for half a century. They were closely related to the Passes, and were like them a fair-skinned, finely built people with quite European features. Language Data on the Yuri language (Jurí) was collected on two occasions in the 19th century, in 1853 and 1867. The american linguist Terrence Kaufman notes that there is good lexical evidence to support a link with Ticuna in a Ticuna–Yurí language family (1994:62, after Nimuendajú 1977:62), though the data has never been explicitly compared (Hammarström 2010). Relation to Carabayo It is commonly assumed that the Juri people and their language has sur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinity
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophical nature of infinity has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including Guillaume de l'Hôpital, l'Hôpital and Johann Bernoulli, Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or Magnitude (mathematics), magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypotheses
A hypothesis (: hypotheses) is a proposed explanation for a phenomenon. A scientific method, scientific hypothesis must be based on observations and make a testable and reproducible prediction about reality, in a process beginning with an educated guess or thought. If a hypothesis is repeatedly independently demonstrated by experiment to be true, it becomes a scientific theory. In colloquial usage, the words "hypothesis" and "theory" are often used interchangeably, but this is incorrect in the context of science. A working hypothesis is a provisionally-accepted hypothesis used for the purpose of pursuing further progress in research. Working hypotheses are frequently discarded, and often proposed with knowledge (and Research ethics, warning) that they are incomplete and thus false, with the intent of moving research in at least somewhat the right direction, especially when scientists are stuck on an issue and brainstorming ideas. A different meaning of the term ''hypothesis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
