Buchholz Psi Functions

	Buchholz Psi Functions Buchholz's psi-functions are a hierarchy of single-argument ordinal functions \psi_\nu(\alpha) introduced by German mathematician Wilfried Buchholz in 1986. These functions are a simplified version of the \theta-functions, but nevertheless have the same strength as those. Later on this approach was extended by Jäger and Schütte. Definition Buchholz defined his functions as follows. Define: Ωξ = ωξ if ξ > 0, Ω0 = 1 The functions ψ''v''(α) for α an ordinal, ''v'' an ordinal at most ω, are defined by induction on α as follows: ψ''v''(α) is the smallest ordinal not in ''C''''v''(α) where ''C''''v''(α) is the smallest set such that ''C''''v''(α) contains all ordinals less than Ω''v'' ''C''''v''(α) is closed under ordinal addition ''C''''v''(α) is closed under the functions ψ''u'' (for ''u''≤ω) applied to arguments less than α. The limit of this notation ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Ordinal Notation In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members of a finite alphabet, to a countable set of ordinals. A Gödel numbering is a function mapping the set of well-formed formulae (a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each well-formed formula with a unique natural number, called its Gödel number. If a Gödel numbering is fixed, then the subset relation on the ordinals induces an ordering on well-formed formulae which in turn induces a well-ordering on the subset of natural numbers. A recursive ordinal notation must satisfy the following two additional properties: # the subset of natural numbers is a recursive set # the induced well-ordering on the subset of natural numbers is a recursive relation There are many such schemes of ordinal notations, including schemes by Wilhelm Acke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Kurt Schütte Kurt Schütte (14 October 1909, Salzwedel – 18 August 1998, Munich) was a Germans, German mathematician who worked on proof theory and ordinal analysis. The Feferman–Schütte ordinal, which he showed to be the precise ordinal bound for predicativity, is named after him. He was the doctoral advisor of 16 students, including Wolfgang Bibel, Wolfgang Maaß, Wolfram Pohlers, and Martin Wirsing. Publications * ** ''Beweistheorie'', Springer, Grundlehren der mathematischen Wissenschaften, 1960; new edition trans. into English as ''Proof Theory'', Springer-Verlag 1977 * ''Vollständige Systeme modaler und intuitionistischer Logik'', Springer 1968 * with Wilfried Buchholz: ''Proof Theory of Impredicative Subsystems of Analysis'', Bibliopolis, Naples 1988 * with Helmut Schwichtenberg''Mathematische Logik'' in Fischer, Hirzebruch et al. (eds.) ''Ein Jahrhundert Mathematik 1890-1990'', Vieweg 1990 References * * External links Kurt Schütte at the Mathematics Genealogy Project ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Takeuti–Feferman–Buchholz Ordinal In the mathematical fields of set theory and proof theory, the Takeuti–Feferman–Buchholz ordinal (TFBO) is a large countable ordinal, which acts as the limit of the range of Buchholz's psi function and Feferman's theta function. It was named by David Madore, after Gaisi Takeuti, Solomon Feferman and Wilfried Buchholz. It is written as \psi_0(\varepsilon_) using Buchholz's psi function, an ordinal collapsing function invented by Wilfried Buchholz, and \theta_(0) in Feferman's theta function, an ordinal collapsing function invented by Solomon Feferman. It is the proof-theoretic ordinal of several formal theories: * \Pi_1^1 -CA + BI, a subsystem of second-order arithmetic * \Pi_1^1-comprehension + transfinite induction * IDω, the system of ω-times iterated inductive definitions Despite being one of the largest large countable ordinals and recursive ordinals, it is still vastly smaller than the proof-theoretic ordinal of ZFC. Definition * Let \Omega_\alpha represent the smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Additively Principal In set theory, a branch of mathematics, an additively indecomposable ordinal ''α'' is any ordinal number that is not 0 such that for any \beta,\gamma<\alpha, we have $\beta+\gamma<\alpha.$ Additively indecomposable ordinals are also called ''gamma numbers'' or ''additive principal numbers''. The additively indecomposable ordinals are precisely those ordinals of the form $\omega^\beta$ for some ordinal $\beta$ . From the continuity of addition in its right argument, we get that if $\beta < \alpha$ and ''α'' is additively indecomposable, then $\beta + \alpha = \alpha.$ Obviously 1 is additively indecomposable, since $0+0<1.$ No finite ordinal other than $1$ is additively indecomposable. Also, $\omega$ is additively indecomposable, since the sum of two finit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Ordinal Notation In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members of a finite alphabet, to a countable set of ordinals. A Gödel numbering is a function mapping the set of well-formed formulae (a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each well-formed formula with a unique natural number, called its Gödel number. If a Gödel numbering is fixed, then the subset relation on the ordinals induces an ordering on well-formed formulae which in turn induces a well-ordering on the subset of natural numbers. A recursive ordinal notation must satisfy the following two additional properties: # the subset of natural numbers is a recursive set # the induced well-ordering on the subset of natural numbers is a recursive relation There are many such schemes of ordinal notations, including schemes by Wilhelm Acke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Binary Relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elements in and in . It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element is ''related'' to an element , if and only if the pair belongs to the set of ordered pairs that defines the ''binary relation''. A binary relation is the most studied special case of an Finitary relation, -ary relation over sets , which is a subset of the Cartesian product X_1 \times \cdots \times X_n. An example of a binary relation is the "divides" relation over the set of prime numbers \mathbb and the set of integers \mathbb, in which each prime is related to each integer that is a Divisibility, multiple of , but not to an integer that is not a multiple of . In this relation, for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]