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Lawvere's Fixed-point Theorem
In mathematics, Lawvere's fixed-point theorem is an important result in category theory. It is a broad abstract generalization of many diagonal arguments in mathematics and logic, such as Cantor's diagonal argument, Cantor's theorem, Russell's paradox, Gödel's first incompleteness theorem, Turing's solution to the Entscheidungsproblem, and Tarski's undefinability theorem. It was first proven by William Lawvere in 1969. Statement Lawvere's theorem states that, for any Cartesian closed category \mathbf and given an object B in it, if there is a weakly point-surjective morphism f from some object A to the exponential object B^A, then every endomorphism g: B \rightarrow B has a fixed point. That is, there exists a morphism b : 1 \rightarrow B (where 1 is a terminal object in \mathbf ) such that g \circ b = b. Applications The theorem's contrapositive is particularly useful in proving many results. It states that if there is an object B in the category such that there i ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism. The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as wel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rice's Theorem
In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable. A semantic property is one about the program's behavior (for instance, does the program terminate for all inputs), unlike a syntactic property (for instance, does the program contain an if-then-else statement). A property is ''non-trivial'' if it is neither true for every partial computable function, nor false for every partial computable function. Rice's theorem can also be put in terms of functions: for any non-trivial property of partial functions, no general and effective method can decide whether an algorithm computes a partial function with that property. Here, a property of partial functions is called ''trivial'' if it holds for all partial computable functions or for none, and an effective decision method is called ''general'' if it decides correctly for every algorithm. The theorem is named after Henry Gordon Rice, who proved it in his doctoral dissertat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleene's Recursion Theorem
In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first proved by Stephen Kleene in 1938 and appear in his 1952 book ''Introduction to Metamathematics''. A related theorem, which constructs fixed points of a computable function, is known as Rogers's theorem and is due to Hartley Rogers, Jr. The recursion theorems can be applied to construct fixed points of certain operations on computable functions, to generate quines, and to construct functions defined via recursive definitions. Notation The statement of the theorems refers to an admissible numbering \varphi of the partial recursive functions, such that the function corresponding to index e is \varphi_e. If F and G are partial functions on the natural numbers, the notation F \simeq G indicates that, for each ''n'', either F(n) and G(n) are both defined and are equal, or else F(n) and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Löb's Paradox
Curry's paradox is a paradox in which an arbitrary claim ''F'' is proved from the mere existence of a sentence ''C'' that says of itself "If ''C'', then ''F''", requiring only a few apparently innocuous logical deduction rules. Since ''F'' is arbitrary, any logic having these rules allows one to prove everything. The paradox may be expressed in natural language and in various logics, including certain forms of set theory, lambda calculus, and combinatory logic. The paradox is named after the logician Haskell Curry. It has also been called Löb's paradox after Martin Hugo Löb, due to its relationship to Löb's theorem. In natural language Claims of the form "if A, then B" are called conditional claims. Curry's paradox uses a particular kind of self-referential conditional sentence, as demonstrated in this example: Even though Germany does not border China, the example sentence certainly is a natural-language sentence, and so the truth of that sentence can be analyzed. The pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonal Lemma
In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence of self-referential sentences in certain formal theories of the natural numbers—specifically those theories that are strong enough to represent all computable functions. The sentences whose existence is secured by the diagonal lemma can then, in turn, be used to prove fundamental limitative results such as Gödel's incompleteness theorems and Tarski's undefinability theorem. Background Let \mathbb be the set of natural numbers. A first-order theory T in the language of arithmetic ''represents'' the computable function f: \mathbb\rightarrow\mathbb if there exists a "graph" predicate \mathcal_f(x, y) in the language of T such that for each n \in \mathbb : \vdash_\,(\forall y) ^\circ f(n)=y) \Leftrightarrow \mathcal_f(^\circ n,\,y)/math> Here ^\circ n is the numeral corresponding to the natural number n, which is defined to b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contrapositive
In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a statement has its antecedent and consequent inverted and flipped. Conditional statement P \rightarrow Q. In formulas: the contrapositive of P \rightarrow Q is \neg Q \rightarrow \neg P . If ''P'', Then ''Q''. — If not ''Q'', Then not ''P''. ''"''If ''it is raining,'' then ''I wear my coat" —'' "If ''I don't wear my coat,'' then ''it isn't raining."'' The law of contraposition says that a conditional statement is true if, and only if, its contrapositive is true. The contrapositive ( \neg Q \rightarrow \neg P ) can be compared with three other statements: ;Inversion (the inverse), \neg P \rightarrow \neg Q:"If ''it is not raining,'' then ''I don't wear my coat''." Unlike the contrapositive, the inverse's truth value is not at all de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial And Terminal Objects
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group is a group homomorphism . In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set ''S'' to itself. In any category, the composition of any two endomorphisms of is again an endomorphism of . It follows that the set of all endomorphisms of forms a monoid, the full transformation monoid, and denoted (or to emphasize the category ). Automorphisms An invertible endomorphism of is called an automorphism. The set of all automorphisms is a subset of with a group structure, called the automorphism group of and denoted . In the following diagram, the arrows denote implication: Endomorphism rings Any two endomorphisms of an abelian group, , can be added togeth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Object
In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined products may still have an exponential law. Definition Let \mathbf be a category, let Z and Y be objects of \mathbf, and let \mathbf have all binary products with Y. An object Z^Y together with a morphism \mathrm\colon (Z^Y \times Y) \to Z is an ''exponential object'' if for any object X and morphism g \colon X\times Y \to Z there is a unique morphism \lambda g\colon X\to Z^Y (called the ''transpose'' of g) such that the following diagram commutes: This assignment of a unique \lambda g to each g establishes an isomorphism (bijection) of hom-sets, \mathrm(X\times Y,Z) \cong \mathrm(X,Z^Y). If Z^Yexists for all objects Z, Y in \mathbf, then the functor (-)^Y \colon \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point-surjective Morphism
In category theory, a point-surjective morphism is a morphism f : X \rightarrow Y that "behaves" like Surjection, surjections on the category of sets. The notion of point-surjectivity is an important one in Lawvere's fixed-point theorem, and it first was introduced by William Lawvere in his original article. Definition Point-surjectivity In a category \mathbf with a terminal object 1, a morphism f : X \rightarrow Y is said to be point-surjective if for every morphism y : 1 \rightarrow Y, there exists a morphism x : 1 \rightarrow X such that f \circ x = y. Weak point-surjectivity If Y is an exponential object of the form B^A for some objects A, B in \mathbf, a weaker (but technically more cumbersome) notion of point-surjectivity can be defined. A morphism f : X \rightarrow B^A is said to be ''weakly'' point-surjective if for every morphism g : A \rightarrow B there exists a morphism x : 1 \rightarrow X such that, for every morphism a : 1 \rightarrow A, we have : \ ... [...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 com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
