Well-founded

	Well-founded In mathematics, a binary relation is called well-founded (or wellfounded or foundational) on a set (mathematics), set or, more generally, a Class (set theory), class if every non-empty subset has a minimal element with respect to ; that is, there exists an such that, for every , one does not have . In other words, a relation is well-founded if: (\forall S \subseteq X)\; [S \neq \varnothing \implies (\exists m \in S) (\forall s \in S) \lnot(s \mathrel m)]. Some authors include an extra condition that is Set-like relation, set-like, i.e., that the elements less than any given element form a set. Equivalently, assuming the axiom of dependent choice, a relation is well-founded when it contains no infinite descending chains, meaning there is no infinite sequence of elements of such that for every natural number . In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then it is called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Axiom Of Regularity In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every Empty set, non-empty Set (mathematics), set ''A'' contains an element that is Disjoint sets, disjoint from ''A''. In first-order logic, the axiom reads: \forall x\,(x \neq \varnothing \rightarrow (\exists y \in x) (y \cap x = \varnothing)). The axiom of regularity together with the axiom of pairing implies that Russell paradox, no set is an element of itself, and that there is no infinite sequence (a_n) such that a_ is an element of a_i for all i. With the axiom of dependent choice (which is a weakened form of the axiom of choice), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, in this context the axiom of regularity is equivalent to the sentence that there are no downward infinite membership chains. The axiom was originally formulated by von Neumann; it was adopted in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Ascending Chain Condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings. These conditions played an important role in the development of the structure theory of commutative rings in the works of David Hilbert, Emmy Noether, and Emil Artin. The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler. Definition A partially ordered set (poset) ''P'' is said to satisfy the ascending chain condition (ACC) if no infinite strictly ascending sequence : a_1 < a_2 < a_3 < \cdots of elements of ''P'' exists. Equivalently, every weakly ascending sequence : $a_1 \leq a_2 \leq a_3 \leq \cdots,$ of elements of ''P'' eventually stabilizes, meaning that there exists a pos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Complete Induction Mathematical induction is a method for proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots all hold. This is done by first proving a simple case, then also showing that if we assume the claim is true for a given case, then the next case is also true. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case n = k, ''then'' it must also hold for the next case n = k + 1. These two steps establish that the statement holds for every natural number n. The base case does not necessarily begin with n = 0, but often with n = 1, and possibly with any fixed natural number n = N, establishing the truth of the statement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Mathematical Induction Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots all hold. This is done by first proving a simple case, then also showing that if we assume the claim is true for a given case, then the next case is also true. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case n = k, ''then'' it must also hold for the next case n = k + 1. These two steps establish that the statement holds for every natural number n. The base case does not necessarily begin with n = 0, but often with n = 1, and possibly with any fixed natural number n = N, establishing the trut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Zermelo–Fraenkel Set Theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Well-order In mathematics, a well-order (or well-ordering or well-order relation) on a set is a total ordering on with the property that every non-empty subset of has a least element in this ordering. The set together with the ordering is then called a well-ordered set (or woset). 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 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 . There may be elements, besides the least element, that have no predecessor (see below for an example). A well-ordered set contains for every subset with an upper bound a least upper bound, namely the least element of the subset of all upper bounds of in . If ≤ is a non-strict well ordering, then < is a st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Transfinite Recursion Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for all ordinals \alpha. Suppose that whenever P(\beta) is true for all \beta < \alpha, then $P(\alpha)$ is also true. Then transfinite induction tells us that $P$ is true for all ordinals. Usually the proof is broken down into three cases: * Zero case: Prove that $P(0)$ is true. * Successor case: Prove that for any successor ordinal $\alpha+1$ , $P(\alpha+1)$ follows from $P(\alpha)$ (and, if necessary, $P(\beta)$ for all $\beta < \alpha$ ). * Limit case: Prove that for any [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Emmy Noether Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which are fundamental in mathematical physics. Noether was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important List of women in mathematics, woman in the history of mathematics. Transcribeonlineat the MacTutor History of Mathematics Archive. As one of the leading mathematicians of her time, she developed theories of ring (mathematics), rings, field (mathematics), fields, and algebras. In physics, Noether's theorem explains the connection between Symmetry (physics), symmetry and conservation laws. in . Noether was born to a Jewish family in the Franconian town of Erlangen; her father was the mathematician Max Noether. She originally planned to teach French and English after passin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Order Theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems in general (although one usually is also interested in the actual difference of two numbers, which is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Transfinite Induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for all ordinals \alpha. Suppose that whenever P(\beta) is true for all \beta < \alpha, then $P(\alpha)$ is also true. Then transfinite induction tells us that $P$ is true for all ordinals. Usually the proof is broken down into three cases: * Zero case: Prove that $P(0)$ is true. * Successor case: Prove that for any successor ordinal $\alpha+1$ , $P(\alpha+1)$ follows from $P(\alpha)$ (and, if necessary, $P(\beta)$ for all $\beta < \alpha$ ). * Limit case: Prove that for any [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Total Order In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a ( strongly connected, formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but generally refers to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Set Theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]