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Computable Ordinal
In mathematics, specifically computability and set theory, an ordinal \alpha is said to be computable or recursive if there is a computable well-ordering of a subset of the natural numbers having the order type \alpha. It is easy to check that \omega is computable. The successor of a computable ordinal is computable, and the set of all computable ordinals is closed downwards. The supremum of all computable ordinals is called the Church–Kleene ordinal, the first nonrecursive ordinal, and denoted by \omega^_1. The Church–Kleene ordinal is a limit ordinal. An ordinal is computable if and only if it is smaller than \omega^_1. Since there are only countably many computable relations, there are also only countably many computable ordinals. Thus, \omega^_1 is countable. The computable ordinals are exactly the ordinals that have an ordinal notation in Kleene's \mathcal. See also *Arithmetical hierarchy *Large countable ordinal *Ordinal analysis *Ordinal notation References * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonrecursive Ordinals
In mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using ordinal collapsing functions. The Church–Kleene ordinal and variants The smallest non-recursive ordinal is the Church Kleene ordinal, \omega_1^, named after Alonzo Church and S. C. Kleene; its order type is the set of all recursive ordinals. Since the successor of a recursive ordinal is recursive, the Church–Kleene ordinal is a limit ordinal. It is also the smallest ordinal that is not hyperarithmetical, and the smallest admissible ordinal after (an ordinal ''α'' is called admissible if L_\alpha \models \mathsf.). The \omega_1^-recursive subsets of are exactly the \Delta^1_1 subsets of .D. MadoreA Zoo of Ordinals(2017). Accessed September 2021. The notation \omega_1^ is in reference to , the first uncountable ordinal, which is the set of all countable ordinals, analogously to how the Church-Kleene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerald Sacks
Gerald Enoch Sacks (1933 – October 4, 2019) was a logician whose most important contributions were in recursion theory. Named after him is Sacks forcing, a forcing notion based on perfect sets and the Sacks Density Theorem, which asserts that the partial order of the recursively enumerable Turing degrees is dense. Sacks had a joint appointment as a professor at the Massachusetts Institute of Technology and at Harvard University starting in 1972 and became emeritus at M.I.T. in 2006 and at Harvard in 2012. Sacks was born in Brooklyn in 1933. He earned his Ph.D. in 1961 from Cornell University under the direction of J. Barkley Rosser, with his dissertation ''On Suborderings of Degrees of Recursive Insolvability''. Among his notable students are Lenore Blum, Harvey Friedman, Sy Friedman, Leo Harrington, Richard Shore, Steve Simpson and Theodore Slaman Theodore Allen Slaman (born April 17, 1954) is a professor of mathematics at the University of California, Berkeley who works in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hartley Rogers Jr
Hartley may refer to: Places Australia *Hartley, New South Wales * Hartley, South Australia **Electoral district of Hartley, a state electoral district Canada *Hartley Bay, British Columbia United Kingdom * Hartley, Cumbria * Hartley, Plymouth, Devon *Hartley Wespall, Hampshire *Hartley, Sevenoaks, Kent * Hartley, Tunbridge Wells, Kent *Hartley, Northumberland (Old Hartley), part of Seaton Sluice *New Hartley, Northumberland United States * Hartley, California *Hartley, Iowa *Hartley, Michigan * Hartley, South Dakota *Hartley, Texas *Hartley County, Texas *Brohard, West Virginia, also Hartley Zimbabwe *Chegutu, formerly Hartley People * Hartley (surname) * Hartley Burr Alexander, (1873–1939), American philosopher * Hartley Alleyne (born 1957), Barbadian cricketer * Hartley Booth (born 1946), British politician * Hartley Coleridge (1796–1849), English writer * Hartley Craig (1917–2007), Australian cricketer * Hartley Douglas Dent (1929–1993), Canadian politici ... [...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] |
Ordinal Analysis
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory. History The field of ordinal analysis was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε0. See Gentzen's consistency proof. Definition Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient portion of arithmetic to make statements about ordinal notations. The proof-theoretic ordinal of such a theory T is the supremum of the order types of all ordinal notations (necessarily recursive, see next section) that the theory can prove are well founded—the supremum of all ordinals \alpha for which the ... [...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] |
Arithmetical Hierarchy
In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy is important in recursion theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic. The Tarski–Kuratowski algorithm provides an easy way to get an upper bound on the classifications assigned to a formula and the set it defines. The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additional formulas and sets. The arithmetical hierarchy of formulas The arithmetical hierarchy assigns classifications to the formulas in the language of first-order arithmetic. The classifications are denoted \Sigma^0_n and \Pi^0_n for natural numbers ''n'' (inclu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleene's O
In set theory and computability theory, Kleene's \mathcal is a canonical subset of the natural numbers when regarded as ordinal notations. It contains ordinal notations for every computable ordinal, that is, ordinals below Church–Kleene ordinal, \omega_1^. Since \omega_1^ is the first ordinal not representable in a computable system of ordinal notations the elements of \mathcal can be regarded as the canonical ordinal notations. Kleene (1938) described a system of notation for all computable ordinals (those less than the Church–Kleene ordinal). It uses a subset of the natural numbers instead of finite strings of symbols. Unfortunately, there is in general no effective way to tell whether some natural number represents an ordinal, or whether two numbers represent the same ordinal. However, one can effectively find notations which represent the ordinal sum, product, and power (see ordinal arithmetic) of any two given notations in Kleene's \mathcal; and given any notation for an o ... [...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] |
Countable Set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (its number of elements) is not greater than that of the natural numbers. A countable set that is not finite is said countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defined here are quite co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β 0, are limits of limits, etc. Properties The classes of successor ordinals and limit ordinals (of various cofinalities) as well as zero exhaust the entire class of ordinals, so these cases are often used in proofs by transfinite induction or definitions by transfinite recursion. Limit ordinals represent a sort of "turning point" in such procedures, in which one must use limiting operations such as taking the union over all preceding ordinals. In principle, one could do anything at limit ordinals, but taking the union is continuous in the order topology and this is usually desirable. If we use the von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal (and this is a fitting obs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
