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Hyperarithmetical Theory
In computability theory, hyperarithmetic theory is a generalization of Turing computability. It has close connections with definability in second-order arithmetic and with weak systems of set theory such as Kripke–Platek set theory. It is an important tool in effective descriptive set theory. The central focus of hyperarithmetic theory is the sets of natural numbers known as hyperarithmetic sets. There are three equivalent ways of defining this class of sets; the study of the relationships between these different definitions is one motivation for the study of hyperarithmetical theory. Hyperarithmetical sets and definability The first definition of the hyperarithmetic sets uses the analytical hierarchy. A set of natural numbers is classified at level \Sigma^1_1 of this hierarchy if it is definable by a formula of second-order arithmetic with only existential set quantifiers and no other set quantifiers. A set is classified at level \Pi^1_1 of the analytical hierarchy if it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computability Theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definable set, definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: * What does it mean for a function (mathematics), function on the natural numbers to be computable? * How can noncomputable functions be classified into a hierarchy based on their level of noncomputability? Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of computational complexity theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing Degree
In computer science and mathematical logic the Turing degree (named after Alan Turing) or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set. Overview The concept of Turing degree is fundamental in computability theory, where sets of natural numbers are often regarded as decision problems. The Turing degree of a set is a measure of how difficult it is to solve the decision problem associated with the set, that is, to determine whether an arbitrary number is in the given set. Two sets are Turing equivalent if they have the same level of unsolvability; each Turing degree is a collection of Turing equivalent sets, so that two sets are in different Turing degrees exactly when they are not Turing equivalent. Furthermore, the Turing degrees are partially ordered, so that if the Turing degree of a set ''X'' is less than the Turing degree of a set ''Y'', then any (possibly noncomputable) procedure that correctly decides whether ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alpha Recursion Theory
In recursion theory, α recursion theory is a generalisation of recursion theory to subsets of admissible ordinals \alpha. An admissible set is closed under \Sigma_1(L_\alpha) functions, where L_\xi denotes a rank of Godel's constructible hierarchy. \alpha is an admissible ordinal if L_ is a model of Kripke–Platek set theory. In what follows \alpha is considered to be fixed. Definitions The objects of study in \alpha recursion are subsets of \alpha. These sets are said to have some properties: *A set A\subseteq\alpha is said to be \alpha-recursively-enumerable if it is \Sigma_1 definable over L_\alpha, possibly with parameters from L_\alpha in the definition. *A is \alpha-recursive if both A and \alpha \setminus A (its relative complement in \alpha) are \alpha-recursively-enumerable. It's of note that \alpha-recursive sets are members of L_ by definition of L. *Members of L_\alpha are called \alpha-finite and play a similar role to the finite numbers in classical recursion th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baire Space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of Baire spaces. The Baire category theorem combined with the properties of Baire spaces has numerous applications in topology, geometry, and analysis, in particular functional analysis. For more motivation and applications, see the article Baire category theorem. The current article focuses more on characterizations and basic properties of Baire spaces per se. Bourbaki introduced the term "Baire space" in honor of René Baire, who investigated the Baire category theorem in the context of Euclidean space \R^n in his 1899 thesis. Definition The definition that follows is based on the notions of meagre (or first category) set (namely, a set that is a countable union of sets whose closure has empty interior) and nonmeagre ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursive 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 computable 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 In proof theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Many-one Reduction
In computability theory and computational complexity theory, a many-one reduction (also called mapping reduction) is a reduction that converts instances of one decision problem (whether an instance is in L_1) to another decision problem (whether an instance is in L_2) using a computable function. The reduced instance is in the language L_2 if and only if the initial instance is in its language L_1. Thus if we can decide whether L_2 instances are in the language L_2, we can decide whether L_1 instances are in the language L_1 by applying the reduction and solving for L_2. Thus, reductions can be used to measure the relative computational difficulty of two problems. It is said that L_1 reduces to L_2 if, in layman's terms L_2 is at least as hard to solve as L_1. This means that any algorithm that solves L_2 can also be used as part of a (otherwise relatively simple) program that solves L_1. Many-one reductions are a special case and stronger form of Turing reductions. With many-one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tarski's Indefinability Theorem
Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that "arithmetical truth cannot be defined in arithmetic". The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system. History In 1931, Kurt Gödel published the incompleteness theorems, which he proved in part by showing how to represent the syntax of formal logic within first-order arithmetic. Each expression of the formal language of arithmetic is assigned a distinct number. This procedure is known variously as Gödel numbering, ''coding'' and, more generally, as arithmetization. In particular, various ''sets'' of expressions are coded as sets of numbers. For various syntactic properties (such as ''being a formula'', ''being a sentence'', etc.), these ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Axioms
In mathematical logic, the Peano axioms (, ), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The axiomatization of arithmetic provided by Peano axioms is commonly called Peano arithmetic. The importance of formalizing arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetical Set
In mathematical logic, an arithmetical set (or arithmetic set) is a set of natural numbers that can be defined by a formula of first-order Peano arithmetic. The arithmetical sets are classified by the arithmetical hierarchy. The definition can be extended to an arbitrary countable set ''A'' (e.g. the set of ''n''-tuples of integers, the set of rational numbers, the set of formulas in some formal language, etc.) by using Gödel numbers to represent elements of the set and declaring a subset of ''A'' to be arithmetical if the set of corresponding Gödel numbers is arithmetical. A function f:A \subseteq \mathbb^k \to \mathbb is called arithmetically definable if the graph of f is an arithmetical set. A real number is called arithmetical if the set of all smaller rational numbers is arithmetical. A complex number is called arithmetical if its real and imaginary parts are both arithmetical. Formal definition A set ''X'' of natural numbers is arithmetical or arithmetically defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type Theory
In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of mathematics. Two influential type theories that have been proposed as foundations are: * Typed λ-calculus of Alonzo Church * Intuitionistic type theory of Per Martin-Löf Most computerized proof-writing systems use a type theory for their foundation. A common one is Thierry Coquand's Calculus of Inductive Constructions. History Type theory was created to avoid paradoxes in naive set theory and formal logic, such as Russell's paradox which demonstrates that, without proper axioms, it is possible to define the set of all sets that are not members of themselves; this set both contains itself and does not contain itself. Between 1902 and 1908, Bertrand Russell proposed various solutions to this problem. By 1908, Russell arrive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lévy Hierarchy
In set theory and mathematical logic, the Lévy hierarchy, introduced by Azriel Lévy in 1965, is a hierarchy of formulas in the formal language of the Zermelo–Fraenkel set theory, which is typically called just the language of set theory. This is analogous to the arithmetical hierarchy, which provides a similar classification for sentences of the language of arithmetic. Definitions In the language of set theory, atomic formulas are of the form x = y or x ∈ y, standing for equality and set membership predicates, respectively. The first level of the Lévy hierarchy is defined as containing only formulas with no unbounded quantifiers and is denoted by \Delta_0=\Sigma_0=\Pi_0.Walicki, Michal (2012). ''Mathematical Logic'', p. 225. World Scientific Publishing Co. Pte. Ltd. The next levels are given by finding a formula in prenex normal form which is provably equivalent over ZFC, and counting the number of changes of quantifiers:T. Jech, 'Set Theory: The Third Millennium Edition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
