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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oswald Veblen
Oswald Veblen (June 24, 1880 – August 10, 1960) was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in .... He proved the Jordan curve theorem in 1905; while this was long considered the first rigorous proof of the theorem, many now also consider Camille Jordan's original proof rigorous. Early life Veblen was born in Decorah, Iowa. His parents were Andrew Anderson Veblen (1848–1932), Professor of Physics at the University of Iowa, and Kirsti (Hougen) Veblen (1851–1908). Veblen's uncle was Thorstein Veblen, noted economist and sociologist. Oswald went to school in Iowa City. He did his undergraduate studies at the University of Iowa, where he received a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Unfortunately 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 φ0 is any normal function, then for any non-zero ordinal α, φ ...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 some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 function 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 ...s: \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 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, φα is the function enumerating the common fixed points of φβ for β<α. These functions are all normal. The Veblen hierarchy In the special case when φ0(α)=ωα this family of functions is known as the Veblen hierarchy. The function φ1 is the same as the ε function: φ1(α)= εα. If then .M. Rathjen[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Collapsing Function
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 than the one being defined, perhaps even large cardinals (though they can be replaced with recursively large ordinals at the cost of extra technical difficulty), and then "collapse" them down to a system of notations for the sought-after ordinal. For this reason, ordinal collapsing functions are described as an impredicative manner of naming ordinals. The details of the definition of ordinal collapsing functions vary, and get more complicated as greater ordinals are being defined, but the typical idea is that whenever the notation system "runs out of fuel" and cannot name a certain ordinal, a much larger ordinal is brought "from above" to give a name to that critical point. An example of how this works will be detailed below, for an ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kruskal's Theorem
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. History The theorem was conjectured by Andrew Vázsonyi and proved by ; a short proof was given by . It has since become a prominent example in reverse mathematics as a statement that cannot be proved within ATR0 (a form of arithmetical transfinite recursion), and a finitary application of the theorem gives the existence of the fast-growing TREE function. In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function. Statement The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree with a root, and given vertices , , call a successor of if the unique path from the root ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rooted Tree
In graph theory, a tree is an undirected graph in which any two vertices are connected by ''exactly one'' path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rewrite Order
In theoretical computer science, in particular in automated reasoning about formal equations, reduction orderings are used to prevent endless loops. Rewrite orders, and, in turn, rewrite relations, are generalizations of this concept that have turned out to be useful in theoretical investigations. Motivation Intuitively, a reduction order ''R'' relates two formal expressions ''s'' and ''t'' if ''t'' is properly "simpler" than ''s'' in some sense. For example, simplification of terms may be a part of a computer algebra program, and may be using the rule set . In order to prove impossibility of endless loops when simplifying a term, the reduction order defined by "''sRt'' if expression ''t'' is properly shorter than expression ''s''" can be used; applying any rule from the set will always properly shorten the term. In contrast, to establish termination of "distributing-out" using the rule ''x''*(''y''+''z'') → ''x''*''y''+''x''*''z'', a more elaborate reduction order will be ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
