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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 ordinal number, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kurt Schütte
Kurt Schütte (14 October 1909 – 18 August 1998) was a 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 Schütteat the Mathematics Genealogy Project The Mathematics Genealogy Pr ... [...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 Definition * Let \Omega_\alpha represent the smallest uncountable ordinal with cardinality \aleph_\alpha. * Let \varepsilon_\beta represent the \betath epsilon number In mathematics, the epsilon ... [...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 Additively indecomposable ordinals were named the ''gamma numbers'' by Cantor,A. Rhea, The Ordinals as a Consummate Abstraction of Number Systems (2017), preprint.p.20 and are also called ''additive principal numbers''. The of additively indecomposable ordinals may be denoted , from the German "Hauptzahl".W. Pohlers, "A short course in ordinal analysis", pp. 27–78. Appearing in [...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 ''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 then .M. Rathjen[...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 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] |
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] |
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] |
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 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 ordinal collapsing functions. 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 * * Ordinal numbers {{Number-stub ... [...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 ordinal number, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Relation
In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs (x, y), where x is an element of X and y is an element of Y. It encodes the common concept of relation: an element x is ''related'' to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation. 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 p is related to each integer z that is a Divisibility, multiple of p, but not to an integer that is not a Multiple (mathematics), multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as -4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13. Binary relations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
