Veblen Ordinal (other)
In mathematics, the Veblen ordinal is either of two large countable ordinals: *The small Veblen ordinal *The large Veblen ordinal See also *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 φ0 is any normal function, then for any non-zero ordinal α, φ ...

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 ω1CK (not to be confused with the first uncountable ordinal, ω1), described below. Ordinal numbers below ω1CK are the recursive ordinals (see below). Countable ordinals larger than this may ...
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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 by ...
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Large Veblen Ordinal
In mathematics, the large Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. 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 functions to produce countable ordinals even for uncountable arguments, and some of which are ordinal collapsing functions In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger t .... The large Veblen ordinal is sometimes denoted by \phi_(0) or \theta(\Omega^\Omega) or \psi(\Omega^). It was constructed by Veblen using an extension of Veblen functions allowing infinitely many arguments. References * * {{Number-stub Ordinal numbers ...
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