|
Disjunction And Existence Properties
In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005). Disjunction property The disjunction property is satisfied by a theory if, whenever a sentence ''A'' ∨ ''B'' is a theorem, then either ''A'' is a theorem, or ''B'' is a theorem. Existence property The existence property or witness property is satisfied by a theory if, whenever a sentence is a theorem, where ''A''(''x'') has no other free variables, then there is some term ''t'' such that the theory proves . Related properties Rathjen (2005) lists five properties that a theory may possess. These include the disjunction property (DP), the existence property (EP), and three additional properties: * The numerical existence property (NEP) states that if the theory proves (\exists x \in \mathbb)\varphi(x), where ''φ'' has no other free variables, then the theory proves \varphi(\ba ... [...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] |
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] |
Intuitionistic Arithmetic
In mathematical logic, Heyting arithmetic is an axiomatization of arithmetic in accordance with the philosophy of intuitionism.Troelstra 1973:18 It is named after Arend Heyting, who first proposed it. Axiomatization As with first-order Peano arithmetic , the intended model of this theory are the natural numbers and the theories characterize addition and multiplication. Heyting arithmetic adopts the axioms of Peano arithmetic, including the signature with zero "0" and the successor "S", but uses intuitionistic logic for inference. In particular, the principle of the excluded middle does not hold in general. Metalogic and theorems As with other theories over intuitionistic logic, various instances of can be proven. For instance, proves equality "=" is decidable for all numbers, :\vdash \forall n. \forall m. \big((n = m)\lor\neg(n = m)\big) In fact, since equality is the only predicate symbol in Heyting arithmetic, it then follows that, for any quantifier-free formula \phi, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursively Enumerable Set
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the set of input numbers for which the algorithm halts is exactly ''S''. Or, equivalently, *There is an enumeration algorithm, algorithm that enumerates the members of ''S''. That means that its output is simply a list of all the members of ''S'': ''s''1, ''s''2, ''s''3, ... . If ''S'' is infinite, this algorithm will run forever. The first condition suggests why the term ''semidecidable'' is sometimes used. More precisely, if a number is in the set, one can ''decide'' this by running the algorithm, but if the number is not in the set, the algorithm runs forever, and no information is returned. A set that is "completely decidable" is a computable set. The second condition suggests why ''computably enumerable'' is used. The abbreviations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harvey Friedman
__NOTOC__ Harvey Friedman (born 23 September 1948)Handbook of Philosophical Logic, , p. 38 is an American mathematical logician at Ohio State University in Columbus, Ohio. He has worked on reverse mathematics, a project intended to derive the axioms of mathematics from the theorems considered to be necessary. In recent years this has advanced to a study of Boolean relation theory, which attempts to justify large cardinal axioms by demonstrating their necessity for deriving certain propositions considered "concrete". Friedman earned his Ph.D. from the Massachusetts Institute of Technology in 1967, with a dissertation on ''Subsystems of Analysis''. His advisor was Gerald Sacks. Friedman received the Alan T. Waterman Award in 1984. He also assumed the title of Vising Scientist at IBM. He delivered the Tarski Lectures in 2007. In 1967, Friedman was listed in the ''Guinness Book of World Records'' for being the world's youngest professor when he taught at Stanford University at age ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products. Definition Let C be a category and let X_1 and X_2 be objects of C. An object is called the coproduct of X_1 and X_2, written X_1 \sqcup X_2, or X_1 \oplus X_2, or sometimes simply X_1 + X_2, if there exist morphisms i_1 : X_1 \to X_1 \sqcup X_2 and i_2 : X_2 \to X_1 \sqcup X_2 satisfying the following universal property: for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \implies g_1 = g_2. Epimorphisms are categorical analogues of onto or surjective functions (and in the category of sets the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts; for example, the inclusion \mathbb\to\mathbb is a ring epimorphism. The dual of an epimorphism is a monomorphism (i.e. an epimorphism in a category ''C'' is a monomorphism in the dual category ''C''op). Many authors in abstract algebra and universal algebra define an epimorphism simply as an ''onto'' or surjective homomorphism. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. In this article, the term "epimorphism" w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous function, continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a Linguistics, linguistic context; see function word. Definition Let ''C'' and ''D'' be category (mathematics), categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Object
In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object. Definition An object P in a category \mathcal is ''projective'' if for any epimorphism e:E\twoheadrightarrow X and morphism f:P\to X, there is a morphism \overline:P\to E such that e\circ \overline=f, i.e. the following diagram commutes: That is, every morphism P\to X factors through every epimorphism E\twoheadrightarrow X. If ''C'' is locally small, i.e., in particular \operatorname_C(P, X) is a set for any object ''X'' in ''C'', this definition is equivalent to the condition that the hom functor (also known as corepresentable functor) : \operatorname(P,-)\colon\mathcal\to\mathbf preserves epimorphisms. Projective objects in abelian categories If the category ''C'' is an abelian category such as, for example, the category of abeli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Terminal Object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): is terminal if for every object in there exists exactly one morphism . Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object. A strict initial object is one for which every morphism into is an isomorphism. Examples * The empty set is the unique initial object in Set, the category of sets. Every one-element set (singleton) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category. * In t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Topos
Free may refer to: Concept * Freedom, having the ability to do something, without having to obey anyone/anything * Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism * Emancipate, to procure political rights, as for a disenfranchised group * Free will, control exercised by rational agents over their actions and decisions * Free of charge, also known as gratis. See Gratis vs libre. Computing * Free (programming), a function that releases dynamically allocated memory for reuse * Free format, a file format which can be used without restrictions * Free software, software usable and distributable with few restrictions and no payment * Freeware, a broader class of software available at no cost Mathematics * Free object ** Free abelian group ** Free algebra ** Free group ** Free module ** Free semigroup * Free variable People * Free (surname) * Free (rapper) (born 1968), or Free Marie, American rapper and media personality ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
