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Grzegorczyk Hierarchy
The Grzegorczyk hierarchy (, ), named after the Polish logician Andrzej Grzegorczyk, is a hierarchy of functions used in computability theory. Every function in the Grzegorczyk hierarchy is a primitive recursive function, and every primitive recursive function appears in the hierarchy at some level. The hierarchy deals with the rate at which the values of the functions grow; intuitively, functions in lower levels of the hierarchy grow slower than functions in the higher levels. Definition First we introduce an infinite set of functions, denoted ''Ei'' for some natural number ''i''. We define : \begin E_0(x,y) & = & x + y \\ E_1(x) & = & x^2 + 2 \\ E_(0) & = & 2 \\ E_(x+1) & = & E_(E_(x)) \\ \end E_0 is the addition function, and E_1 is a unary function which squares its argument and adds two. Then, for each ''n'' greater than 1, E_n(x)=E^_(2), i.e. the ''x''-th iterate of E_ evaluated at 2. From these functions we define the Grzegorczyk hierarchy. \mathcal^n, the ''n''- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Andrzej Grzegorczyk
Andrzej Grzegorczyk (; 22 August 1922 – 20 March 2014) was a Polish logician, mathematician, philosopher and ethicist. He was noted for his work in computability, mathematical logic and the foundations of mathematics. Family In 1953, Grzegorczyk married Renata Maria Grzegorczykowa, a Polish philologist and expert in polonist linguistics. They had a daughter and a son. Grzegorczyk died of natural causes in Warsaw on 20 March 2014 at the age of 91. His body is buried in the Cemetery of Pruszków. See also * List of Polish people Sources *Odintsov, Sergei Pavlovich (2018)Larisa Maksimova on Implication, Interpolation, and Definability Springer International Publishing, Cham *Golińska-Pilarek, Joanna; Huuskonen, Taneli (2017): ''Grzegorczyk's non-Fregean logics and their formal properties''. In Urbaniak, Rafał; Payette, Gillman (editors) (2017): Applications of Formal Philosophy: The Road Less Travelled'. Springer International Publishing, Cham, Chapter 12, pp. 24 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Partition Of A Set
In mathematics, a partition of a set is a grouping of its elements into Empty set, non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a Set (mathematics), set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory. Definition and notation A partition of a set ''X'' is a set of non-empty subsets of ''X'' such that every element ''x'' in ''X'' is in exactly one of these subsets (i.e., the subsets are nonempty mutually disjoint sets). Equivalently, a family of sets ''P'' is a partition of ''X'' if and only if all of the following conditions hold: *The family ''P'' does not contain the empty set (that is \emptyset \notin P). *The union (set theory), union of the sets in ''P'' is equal to ''X'' (that is \textstyle\bigcup_ A = X). The sets in ''P'' are said ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Journal Of Symbolic Logic
The '' Journal of Symbolic Logic'' is a peer-reviewed mathematics journal published quarterly by Association for Symbolic Logic. It was established in 1936 and covers mathematical logic. The journal is indexed by '' Mathematical Reviews'', Zentralblatt MATH, and Scopus. Its 2009 MCQ was 0.28, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... was 0.631. External links * Mathematical logic journals Academic journals established in 1936 Multilingual journals Quarterly journals Association for Symbolic Logic academic journals Logic journals Cambridge University Press academic journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Ordinal Analysis
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory. In addition to obtaining the proof-theoretic ordinal of a theory, in practice ordinal analysis usually also yields various other pieces of information about the theory being analyzed, for example characterizations of the classes of provably recursive, hyperarithmetical, or \Delta^1_2 functions of the theory. History The field of ordinal analysis was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε0. See Gentzen's consistency proof. Definition Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Fast-growing Hierarchy
In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy, or a Schwichtenberg-Wainer hierarchy) is an ordinal-indexed family of rapidly increasing functions ''f''α: N → N (where N is the set of natural numbers , and α ranges up to some large countable Ordinal number, ordinal). A primary example is the Wainer hierarchy, or Löb–Wainer hierarchy, which is an extension to all α < Epsilon numbers (mathematics), ε0. Such hierarchies provide a natural way to classify computable functions according to rate-of-growth and Computational complexity theory, computational complexity. Definition Let μ be a large countable ordinal such that to every limit ordinal α < μ there is assigned a fundamental sequence (ordinals), fundamental sequence (a strictly increasing sequence of ordinals whose supremum is α). A fast-growing hierarchy of functions ''f''α: N ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Stan S
Stan or STAN may refer to: People * Stan (given name), a list of people with the given name ** Stan Laurel (1890–1965), English comic actor, part of duo Laurel and Hardy * Stan (surname), a Romanian surname * Stan! (born 1964), American author, cartoonist and games designer Steven Brown * Stan (singer) (born 1987), Greek singer, born Stratos Antipariotis Arts and entertainment Fictional characters * Stan Marsh, in the animated TV series ''South Park'' * Stan, an alligator in the 2006 Disney animated film ''The Wild'' * Grunkle Stan, in the animated TV series ''Gravity Falls'' * Stan, in the 2009 American fantasy comedy movie '' 17 Again'' * Stan, from the film ''Crawl'' * Stan Beeman, in the TV series ''The Americans'' * Stan Carter, in the British soap opera ''EastEnders'' * Stan Edgar, in the Amazon Prime Video series '' The Boys'' * Stan Gable, in the ''Revenge of the Nerds'' film series played by Ted McGinley * Stan Ogden, in the British soap opera ''Coronation Street'' * Sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Martin Löb
Martin Hugo Löb (; 31 March 1921 – 21 August 2006) was a German mathematician. He settled in the United Kingdom after the Second World War and specialised in mathematical logic. He moved to the Netherlands in the 1970s, where he remained in retirement. He is perhaps best known for having formulated Löb's theorem in 1955. Early life and education Löb grew up in Berlin, but escaped from the Third Reich, arriving in the UK just before the outbreak of the Second World War. As an enemy alien, he was deported on the '' Dunera'' to an internment camp at Hay in Australia in 1940, where the 19-year-old Löb was taught mathematics by other internees. His teacher, Felix Behrend, was later a professor at Melbourne University. Löb was allowed to return to the UK in 1943, and he studied at the University of London after the War. After graduating, he became a research student with Reuben Goodstein at the University of Leicester. He completed his PhD and became an assistant lect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Epsilon Numbers (mathematics)
In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication. The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ''ε'' that satisfy the equation :\varepsilon = \omega^\varepsilon, \, in which ω is the smallest infinite ordinal. The least such ordinal is ''ε''0 (pronounced epsilon nought (chiefly British), epsilon naught (chiefly American), or epsilon zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals: :\varepsilon_0 = \omega^ = \sup \left\\,, where is the supremum, which is equivalent to set union in the case of the von Neumann representation of ordinals. Larger ordinal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Limit Ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β 0, are limits of limits, etc. Properties The classes of successor ordinals and limit ordinals (of various cofinalities) as well as zero exhaust the entire class of ordinals, so these cases are often used in proofs by transfinite induction or definitions by transfinite recursion. Limit ordinals represent a sort of "turning point" in such procedures, in which one must use limiting operations such as taking the union over all preceding ordinals. In principle, one could do anything at limit ordinals, but taking the union is continuous in the order topology and this is usually desirable. If we use the von Neumann cardinal assignment, every infinite cardinal number In mathematics, a cardinal number, or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Fast-growing Hierarchy
In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy, or a Schwichtenberg-Wainer hierarchy) is an ordinal-indexed family of rapidly increasing functions ''f''α: N → N (where N is the set of natural numbers , and α ranges up to some large countable Ordinal number, ordinal). A primary example is the Wainer hierarchy, or Löb–Wainer hierarchy, which is an extension to all α < Epsilon numbers (mathematics), ε0. Such hierarchies provide a natural way to classify computable functions according to rate-of-growth and Computational complexity theory, computational complexity. Definition Let μ be a large countable ordinal such that to every limit ordinal α < μ there is assigned a fundamental sequence (ordinals), fundamental sequence (a strictly increasing sequence of ordinals whose supremum is α). A fast-growing hierarchy of functions ''f''α: N ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Ordinal Number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally using linearly ordered greek letter variables that include the natural numbers and have the property that every set of ordinals has a least or "smallest" element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number \omega (omega) to be the least element that is greater than every natural number, along with ordinal numbers , , etc., which are even greater than . A linear order such that every non-empty subset has a least element is called a well-order. The axiom of choice implies that every set can be well-orde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |