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Analytical Hierarchy
In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers over both the set of natural numbers, \mathbb, and over functions from \mathbb to \mathbb. The analytical hierarchy of sets classifies sets by the formulas that can be used to define them; it is the lightface version of the projective hierarchy. The analytical hierarchy of formulas The notation \Sigma^1_0 = \Pi^1_0 = \Delta^1_0 indicates the class of formulas in the language of second-order arithmetic with number quantifiers but no set quantifiers. This language does not contain set parameters. The Greek letters here are lightface symbols, which indicate this choice of language. Each corresponding boldface symbol denotes the corresponding class of formulas in the extended language with a parameter for each real; see projective hierarch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor Space
In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the" Cantor space. Examples The Cantor set itself is a Cantor space. But the canonical example of a Cantor space is the countably infinite topological product of the discrete 2-point space . This is usually written as 2^\mathbb or 2ω (where 2 denotes the 2-element set with the discrete topology). A point in 2ω is an infinite binary sequence, that is a sequence which assumes only the values 0 or 1. Given such a sequence ''a''0, ''a''1, ''a''2,..., one can map it to the real number :\sum_^\infty \frac. This mapping gives a homeomorphism from 2ω onto the Cantor set, demonstrating that 2ω is indeed a Cantor space. Cantor spaces occur abundantly in real analysis. For example, they exist as subspaces in every perfect, complete metric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Effective Descriptive Set Theory
Effective descriptive set theory is the branch of descriptive set theory dealing with sets of reals having lightface definitions; that is, definitions that do not require an arbitrary real parameter (Moschovakis 1980). Thus effective descriptive set theory combines descriptive set theory with recursion theory. Constructions Effective Polish space An effective Polish space is a complete separable metric space that has a computable presentation. Such spaces are studied in both effective descriptive set theory and in constructive analysis. In particular, standard examples of Polish spaces such as the real line, the Cantor set and the Baire space are all effective Polish spaces. Arithmetical hierarchy The arithmetical hierarchy, arithmetic hierarchy or Kleene– Mostowski hierarchy classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called "arithmetical". More formally, the arithmetical hierarchy assigns clas ... [...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 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 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 subrecursive hierarchies, formal methods, and formal languages. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast-growing Hierarchy
In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy) is an ordinal-indexed family of rapidly increasing functions ''f''α: N → N (where N is the set of natural numbers , and α ranges up to some large countable ordinal). A primary example is the Wainer hierarchy, or Löb–Wainer hierarchy, which is an extension to all α < ε0. Such hierarchies provide a natural way to classify s according to rate-of-growth and . ...
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Analytic Set
In the mathematical field of descriptive set theory, a subset of a Polish space X is an analytic set if it is a continuous image of a Polish space. These sets were first defined by and his student . Definition There are several equivalent definitions of analytic set. The following conditions on a subspace ''A'' of a Polish space ''X'' are equivalent: *''A'' is analytic. *''A'' is empty or a continuous image of the Baire space ωω. *''A'' is a Suslin space, in other words ''A'' is the image of a Polish space under a continuous mapping. *''A'' is the continuous image of a Borel set in a Polish space. *''A'' is a Suslin set, the image of the Suslin operation. *There is a Polish space Y and a Borel set B\subseteq X\times Y such that A is the projection of B; that is, : A=\. *''A'' is the projection of a closed set in the cartesian product of ''X'' with the Baire space. *''A'' is the projection of a Gδ set in the cartesian product of ''X'' with the Cantor space. An alterna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well Ordering
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-order relation is then called a well-ordered set. In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering. Every non-empty well-ordered set has a least element. Every element ''s'' of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than ''s''. There may be elements besides the least element which have no predecessor (see below for an example). A well-ordered set ''S'' contains for every subset ''T'' with an upper bound a least upper bound, namely the least element of the subset of all upper bounds of ''T'' in ''S''. If ≤ is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Constructibility
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible universe, constructible. The axiom is usually written as ''V'' = ''L'', where ''V'' and ''L'' denote the von Neumann universe and the constructible universe, respectively. The axiom, first investigated by Kurt Gödel, is inconsistent with the proposition that zero sharp exists and stronger large cardinal axioms (see list of large cardinal properties). Generalizations of this axiom are explored in inner model theory. Implications The axiom of constructibility implies the axiom of choice (AC), given Zermelo–Fraenkel set theory without the axiom of choice (ZF). It also settles many natural mathematical questions that are independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC); for example, the axiom of constructibility implies the Continuum hypothesis#The generalized continuum hypothesis, generalized continuum hypothesis, the negation of ... [...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. 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 theory. *Members ... [...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 an equivalent formula in prenex normal form, and counting the number of changes of quantifiers: In the theory ZFC, a formula A is called: \Sigma _ if A is equivalent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polish Space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians— Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory. Common examples of Polish spaces are the real line, any separable Banach space, the Cantor space, and the Baire space. Additionally, some spaces that are not complete metric spaces in the usual metric may be Polish; e.g., the open interval (0, 1) is Polish. Between any two uncountable Polish spaces, there is a Borel isomorphism; that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baire Space (set Theory)
In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. This space is commonly used in descriptive set theory, to the extent that its elements are often called "reals". It is denoted NN, ωω, by the symbol \mathcal or also ωω, not to be confused with the countable ordinal obtained by ordinal exponentiation. The Baire space is defined to be the Cartesian product of countably infinitely many copies of the set of natural numbers, and is given the product topology (where each copy of the set of natural numbers is given the discrete topology). The Baire space is often represented using the tree of finite sequences of natural numbers. The Baire space can be contrasted with Cantor space, the set of infinite sequences of binary digits. Topology and trees The product topology used to define the Baire space can be described more concretely in terms of trees. The basic open sets of the product topology are cylinder sets, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |