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Effective Topos
In mathematics, the effective topos introduced by captures the mathematical idea of effectivity within the category theoretical framework. Definition Preliminaries Kleene realizability The topos is based on the partial combinatory algebra given by Kleene's first algebra _1. In Kleene's notion of recursive realizability, any predicate is assigned realizing numbers, i.e. a subset of . The extremal propositions are \top and \bot, realized by and \. However in general, this process assigns more data to a proposition than just a binary truth value. A formula with k free variables will give rise to a map in (\mathcal P)^ the values of which is the subset of corresponding realizers. Realizability topoi is a prime example of a realizability topos. These are a class of elementary topoi with an intuitionistic internal logic and fulfilling a form of dependent choice. They are generally not Grothendieck topoi. In particular, the effective topos is (_1). Other realizability topos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computability
Computability is the ability to solve a problem by an effective procedure. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked to the existence of an algorithm to solve the problem. The most widely studied models of computability are the Turing-computable and μ-recursive functions, and the lambda calculus, all of which have computationally equivalent power. Other forms of computability are studied as well: computability notions weaker than Turing machines are studied in automata theory, while computability notions stronger than Turing machines are studied in the field of hypercomputation. Problems A central idea in computability is that of a (computational) problem, which is a task whose computability can be explored. There are two key types of problems: * A decision problem fixes a set ''S'', which may be a set of strings, natural numbers, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heyting Algebra
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' called ''implication'' such that (''c'' ∧ ''a'') ≤ ''b'' is equivalent to ''c'' ≤ (''a'' → ''b''). From a logical standpoint, ''A'' → ''B'' is by this definition the weakest proposition for which modus ponens, the inference rule ''A'' → ''B'', ''A'' ⊢ ''B'', is sound. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced in 1930 by Arend Heyting to formalize intuitionistic logic. Heyting algebras are distributive lattices. Every Boolean algebra is a Heyting algebra when ''a'' → ''b'' is defined as ¬''a'' ∨ ''b'', as is every complete distributive lattice satisfying a one-sided infinite distributive law when ''a'' → ''b'' is taken to be t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Specker Sequence
In computability theory, a Specker sequence is a computable, monotonically increasing, bounded sequence of rational numbers whose supremum is not a computable real number. The first example of such a sequence was constructed by Ernst Specker (1949). The existence of Specker sequences has consequences for computable analysis. The fact that such sequences exist means that the collection of all computable real numbers does not satisfy the least upper bound principle of real analysis, even when considering only computable sequences. A common way to resolve this difficulty is to consider only sequences that are accompanied by a modulus of convergence; no Specker sequence has a computable modulus of convergence. More generally, a Specker sequence is called a ''recursive counterexample'' to the least upper bound principle, i.e. a construction that shows that this theorem is false when restricted to computable reals. The least upper bound principle has also been analyzed in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intermediate Value Theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two important corollaries: # If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem). # The image of a continuous function over an interval is itself an interval. Motivation This captures an intuitive property of continuous functions over the real numbers: given ''f'' continuous on ,2/math> with the known values f(1) = 3 and f(2) = 5, then the graph of y = f(x) must pass through the horizontal line y = 4 while x moves from 1 to 2. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper. Theorem The intermediate value theorem states the following: Consider the closed interval I = ,b/math> ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrey Markov Jr
Andrey (Андрей) is a masculine given name predominantly used in Slavic languages, including Belarusian, Bulgarian, and Russian. The name is derived from the ancient Greek Andreas (Ἀνδρέας), meaning "man" or "warrior". In Eastern Orthodox Christianity, Andrey holds religious significance, particularly due to Saint Andrew, the patron saint of several countries, whose legacy has contributed to the name’s popularity across Orthodox nations. In Spanish-speaking countries, Andrey can be interpreted as a portmanteau of the name Andrés and '' Rey'', the Spanish word for ''king''. People with the given name * Andrey (footballer, born 1983), Andrey Nazário Afonso, goalkeeper for Avenida * Andrey (footballer, born 1993), Andrey da Silva Ventura, goalkeeper for Sampaio Corrêa * Andrey (footballer, born 1996), Andrey Falinski Rodrigues, midfielder for Betim Futebol * Andrey (footballer, born February 1998), Andrey Ramos do Nascimento, midfielder for Coritiba * Andr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indecomposability (intuitionistic Logic)
In intuitionistic analysis and in computable analysis, indecomposability or indivisibility (, from the adjective ''unzerlegbar'') is the principle that the continuum cannot be partitioned into two nonempty pieces. This principle was established by Brouwer in 1928 English translation of §1 see p.490–492 of: using intuitionistic principles, and can also be proven using Church's thesis. The analogous property in classical analysis is the fact that every continuous function from the continuum to is constant. It follows from the indecomposability principle that any property of real numbers that is ''decided'' (each real number either has or does not have that property) is in fact trivial (either all the real numbers have that property, or else none of them do). Conversely, if a property of real numbers is not trivial, then the property is not decided for all real numbers. This contradicts the law of the excluded middle, according to which every property of the real numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Sequence
In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are less than that given distance from each other. Cauchy sequences are named after Augustin-Louis Cauchy; they may occasionally be known as fundamental sequences. It is not sufficient for each term to become arbitrarily close to the term. For instance, in the sequence of square roots of natural numbers: a_n=\sqrt n, the consecutive terms become arbitrarily close to each other – their differences a_-a_n = \sqrt-\sqrt = \frac d. As a result, no matter how far one goes, the remaining terms of the sequence never get close to ; hence the sequence is not Cauchy. The utility of Cauchy sequences lies in the fact that in a complete metric space (one where all such sequences are known to converge to a limit), the criterion for convergence depends ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independence Of Premise
In proof theory and constructive mathematics, the principle of independence of premise (IP) states that if φ and ∃''x'' θ are sentences in a formal theory and is provable, then is provable. Here ''x'' cannot be a free variable of φ, while θ can be a predicate depending on it. The main application of the principle is in the study of intuitionistic logic, where the principle is not generally valid. Its crucial equivalent special case is discussed below. The principle is valid in classical logic. Discussion As is common, the domain of discourse is assumed to be inhabited. That is, part of the theory is at least some term. For the discussion we distinguish one such term as ''a''. In the theory of the natural numbers, this role may be played by the number ''7''. Below, ''φ'' and ''ψ'' denote propositions not depending on ''x'', while ''θ'' is a predicate that can depend on in. The following is easily established: * Firstly, if φ is established to be true, then if one ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Church%27s Thesis (constructive Mathematics)
In constructive mathematics, Church's thesis is the principle stating that all total functions are computable functions. The similarly named Church–Turing thesis states that every '' effectively calculable function'' is a ''computable function'', thus collapsing the former notion into the latter. is stronger in the sense that with it every function is computable. The constructivist principle is however also given, in different theories and incarnations, as a fully formal axiom. The formalizations depends on the definition of "function" and "computable" of the theory at hand. A common context is recursion theory as established since the 1930's. Adopting as a principle, then for a predicate of the form of a family of existence claims (e.g. \exists! y. \varphi(x,y) below) that is proven not to be validated for all x in a computable manner, the contrapositive of the axiom implies that this is then not validated by ''any'' total function (i.e. no mapping corresponding to x\mapsto y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov's Principle
Markov's principle (also known as the Leningrad principle), named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below. The principle is logically valid classically, but not in intuitionistic constructive mathematics. However, many particular instances of it are nevertheless provable in a constructive context as well. History The principle was first studied and adopted by the Russian school of constructivism, together with choice principles and often with a realizability perspective on the notion of mathematical function. In computability theory In the language of computability theory, Markov's principle is a formal expression of the claim that if it is impossible that an algorithm does not terminate, then for some input it does terminate. This is equivalent to the claim that if a set and its complement are both computably enumerable, then the set is decidable. These statements are provable in cla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
