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Suslin Cardinal
In mathematics, a cardinal λ < Θ is a Suslin cardinal if there exists a set P ⊂ 2ω such that P is λ-Suslin but P is not λ'-Suslin for any λ' < λ. It is named after the n (1894–1919). See also * Suslin representation * |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinal Number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ''transfinite'' cardinal numbers, often denoted using the Hebrew symbol \aleph ( aleph) followed by a subscript, describe the sizes of infinite sets. Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Θ (set Theory)
In set theory, Θ (pronounced like the letter theta) is the least nonzero ordinal α such that there is no surjection from the reals onto α. If the axiom of choice (AC) holds (or even if the reals can be wellordered), then Θ is simply (2^)^+, the cardinal successor of the cardinality of the continuum. However, Θ is often studied in contexts where the axiom of choice fails, such as models of the axiom of determinacy. Θ is also the supremum of the lengths of all prewellorderings of the reals. Proof of existence It may not be obvious that it can be proven, without using AC, that there even exists a nonzero ordinal onto which there is no surjection from the reals (if there is such an ordinal, then there must be a least one because the ordinals are wellordered). However, suppose there were no such ordinal. Then to every ordinal α we could associate the set of all prewellorderings of the reals having length α. This would give an injection from the class of all ordinals into ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
λ-Suslin
In mathematics, a Suslin representation of a set of reals (more precisely, elements of Baire space) is a tree whose projection is that set of reals. More generally, a subset ''A'' of ''κ''ω is ''λ''-Suslin if there is a tree ''T'' on ''κ'' × ''λ'' such that ''A'' = p 'T'' By a tree on ''κ'' × ''λ'' we mean here a subset ''T'' of the union of ''κ''''i'' × ''λ''''i'' for all ''i'' ∈ N (or ''i'' < ω in set-theoretical notation). Here, p 'T''= is the projection of ''T'', where 'T''= is the set of es through ''T''. Since 'T''is a closed set for the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russia
Russia (, , ), or the Russian Federation, is a List of transcontinental countries, transcontinental country spanning Eastern Europe and North Asia, Northern Asia. It is the List of countries and dependencies by area, largest country in the world, with its internationally recognised territory covering , and encompassing one-eighth of Earth's inhabitable landmass. Russia extends across Time in Russia, eleven time zones and shares Borders of Russia, land boundaries with fourteen countries, more than List of countries and territories by land borders, any other country but China. It is the List of countries and dependencies by population, world's ninth-most populous country and List of European countries by population, Europe's most populous country, with a population of 146 million people. The country's capital and List of cities and towns in Russia by population, largest city is Moscow, the List of European cities by population within city limits, largest city entirely within E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Yakovlevich Suslin
Mikhail Yakovlevich Suslin (russian: Михаи́л Я́ковлевич Су́слин; , November 15, 1894 – 21 October 1919, Krasavka) (sometimes transliterated Souslin) was a Russian mathematician who made major contributions to the fields of general topology and descriptive set theory. Biography Mikhail Suslin was born on November 15, 1894 in the village of Krasavka, the only child of poor peasants Yakov Gavrilovich and Matrena Vasil'evna Suslin. From a young age, Suslin showed a keen interest in mathematics and was encouraged to continue his education by his primary school teacher, Vera Andreevna Teplogorskaya-Smirnova. From 1905 to 1913 he attended Balashov boys' grammar school. In 1913, Suslin enrolled at the Imperial Moscow University and studied under the tutelage of Nikolai Luzin. He graduated with a degree in mathematics in 1917 and immediately began working at the Ivanovo-Voznesensk Polytechnic Institute. Suslin died of typhus in the 1919 Moscow epidemic follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Akihiro Kanamori
is a Japanese-born American mathematician. He specializes in set theory and is the author of the monograph on large cardinal property, large cardinals, ''The Higher Infinite''. He has written several essays on the history of mathematics, especially set theory. Kanamori graduated from California Institute of Technology and earned a Ph.D. from the University of Cambridge (King's College, Cambridge, King's College). He is a professor of mathematics at Boston University. With Matthew Foreman he is the editor of the ''Handbook of Set Theory'' (2010). Selected publications * A. Kanamori, Menachem Magidor, M. MagidorThe evolution of large cardinal axioms in set theory in: ''Higher set theory'' (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1977), Lecture Notes in Mathematics, 669, Springer, 99–275. * Robert Solovay, R. M. Solovay, W. N. Reinhardt, A. KanamoriStrong axioms of infinity and elementary embeddings ''Annals of Mathematical Logic'', 13(1978), 73–116. * A. Kan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suslin Representation
In mathematics, a Suslin representation of a set of reals (more precisely, elements of Baire space) is a tree whose projection is that set of reals. More generally, a subset ''A'' of ''κ''ω is ''λ''-Suslin if there is a tree ''T'' on ''κ'' × ''λ'' such that ''A'' = p 'T'' By a tree on ''κ'' × ''λ'' we mean here a subset ''T'' of the union of ''κ''''i'' × ''λ''''i'' for all ''i'' ∈ N (or ''i'' < ω in set-theoretical notation). Here, p 'T''= is the projection of ''T'', where 'T''= is the set of es through ''T''. Since 'T''is a closed set for the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suslin Line
In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent. (Suslin is also sometimes written with the French transliteration as , from the Cyrillic .) Formulation Suslin's problem asks: Given a non-empty totally ordered set ''R'' with the four properties # ''R'' does not have a least nor a greatest element; # the order on ''R'' is dense (between any two distinct elements there is another); # the order on ''R'' is complete, in the sense that every non-empty bounded subset has a supremum and an infimum; and # every collection of mutually disjoint non-empty open intervals in ''R'' is countable (this is the countable chain condition for the order topology of ''R''), is ''R'' necessarily order-isomorphic to the real line R? If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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