Ackermann Ordinal

	Ackermann Ordinal In mathematics, the Ackermann ordinal is a certain large countable ordinal, named after Wilhelm Ackermann. The term "Ackermann ordinal" is also occasionally used for the small Veblen ordinal, a somewhat larger ordinal. There is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen function In mathematics, the Veblen functions are a hierarchy of normal functions ( continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If ''φ''0 is any normal function, then for any non-zero ordinal '' ...s to produce countable ordinals even for uncountable arguments, and some of which are " collapsing functions". The last one is an extension of the Veblen functions for more than 2 arguments. The smaller Ackermann ordinal is the limit of a system of ordinal notations invented by , and is sometimes denot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Large Countable Ordinal In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relevance to proof theory still have computable ordinal notations (see ordinal analysis). However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not (for reasons somewhat analogous to the unsolvability of the halting problem); various more-concrete ways of defining ordinals that definitely have notations are available. Since there are only countably many notations, all ordinals with notations are exhausted well below the first uncountable ordinal ω1; their supremum is called ''Church–Kleene'' ω1 or ω (not to be confused with the first uncountable ordinal, ω1), described below. Ordinal numbers below ω are the ''recursive'' ordinals (see below). Countable ordinals larger than this m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Wilhelm Ackermann Wilhelm Friedrich Ackermann (; ; 29 March 1896 – 24 December 1962) was a German mathematician and logician best known for his work in mathematical logic and the Ackermann function, an important example in the theory of computation. Biography Ackermann was born in Herscheid, Germany, and was awarded a Ph.D. by the University of Göttingen in 1925 for his thesis ''Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit'', which was a consistency proof of arithmetic apparently without Peano induction (although it did use e.g. induction over the length of proofs). This was one of two major works in proof theory in the 1920s and the only one following Hilbert's school of thought. From 1929 until 1948, he taught at the Arnoldinum Gymnasium in Burgsteinfurt, and then at Lüdenscheid until 1961. He was also a corresponding member of the Akademie der Wissenschaften (''Academy of Sciences'') in Göttingen, and was an honorary professor at the U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Small Veblen Ordinal In mathematics, the small Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. It is occasionally called the Ackermann ordinal, though the Ackermann ordinal described by is somewhat smaller than the small Veblen ordinal. There is no standard notation for ordinals beyond the Feferman–Schütte ordinal \Gamma_0. Most systems of notation use symbols such as \psi(\alpha), \theta(\alpha), \psi_\alpha(\beta), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are " collapsing functions". The small Veblen ordinal \theta_(0) or \psi(\Omega^) is the limit of ordinals that can be described using a version of Veblen functions with finitely many arguments. It is the ordinal that measures the strength of Kruskal's theorem. It is also the ordinal type of a certain ordering of rooted tree In graph theory, a tree is an undirected graph in which any two vertices are connected ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Feferman–Schütte Ordinal In mathematics, the Feferman–Schütte ordinal (Γ0) is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion. It is named after Solomon Feferman and Kurt Schütte, the former of whom suggested the name Γ0. There is no standard notation for ordinals beyond the Feferman–Schütte ordinal. There are several ways of representing the Feferman–Schütte ordinal, some of which use ordinal collapsing functions: \psi(\Omega^\Omega), \theta(\Omega), \varphi_\Omega(0), or \varphi(1,0,0). Definition The Feferman–Schütte ordinal can be defined as the smallest ordinal that cannot be obtained by starting with 0 and using the operations of ordinal addition and the Veblen functions ''φ''''α''(''β''). That is, it is the smallest ''α'' such that ''φ''''α''(0) = ''α''. Properties This ordinal is sometimes said to be the first impredicative ordinal, though this is controversial, partly because there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Veblen Function In mathematics, the Veblen functions are a hierarchy of normal functions ( continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If ''φ''0 is any normal function, then for any non-zero ordinal ''α'', ''φ''''α'' is the function enumerating the common fixed points of ''φ''''β'' for ''β''<''α''. These functions are all normal. Veblen hierarchy In the special case when ''φ''₀(''α'')=ω^''α'' this family of functions is known as the Veblen hierarchy. The function ''φ''₁ is the same as the ε function: ''φ''₁(''α'')= ε_''α''. If $\alpha < \beta \,,$ then $\varphi_(\varphi_(\gamma)) = \varphi_(\gamma)$ .M. Rathjen [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]

Veblen hierarchy