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Epsilon-induction
In set theory, \in-induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets satisfy a given property. Considered as an axiomatic principle, it is called the axiom schema of set induction. The principle implies transfinite induction and recursion. It may also be studied in a general context of induction on well-founded relations. Statement The schema is for any given property \psi of sets and states that, if for every set x, the truth of \psi(x) follows from the truth of \psi for all elements of x, then this property \psi holds for all sets. In symbols: :\forall x. \Big(\big(\forall (y \in x). \psi(y)\big)\,\to\,\psi(x)\Big)\,\to\,\forall z. \psi(z) Note that for the "bottom case" where x denotes the empty set \, the subexpression \forall(y\in x).\psi(y) is vacuously true for all propositions and so that implication is proven by just proving \psi(\). In words, if a property is persistent when collecting any sets with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Set Theory
Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "=" and "\in" of classical set theory is usually used, so this is not to be confused with a constructive types approach. On the other hand, some constructive theories are indeed motivated by their interpretability in type theories. In addition to rejecting the principle of excluded middle (), constructive set theories often require some logical quantifiers in their axioms to be bounded, motivated by results tied to impredicativity. Introduction Constructive outlook Use of intuitionistic logic The logic of the set theories discussed here is constructive in that it rejects , i.e. that the disjunction \phi \lor \neg \phi automatically holds for all propositions. As a rule, to prove the excluded middle for a proposition P, i.e. to prove the particular disjunction P \lor \neg P, either P or \neg P needs to be explicitly prov ... [...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 is also true. Then transfinite induction tells us that is true for all ordinals. Usually the proof is broken down into three cases: * Zero case: Prove that is true. * Successor case: Prove that for any successor ordinal , follows from (and, if necessary, for all ). * Limit case: Prove that for any [...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 non-empty set ''A'' contains an element that is disjoint from ''A''. In first-order logic, the axiom reads: : \forall x\,(x \neq \varnothing \rightarrow \exists y(y \in x\ \land y \cap x = \varnothing)). The axiom of regularity together with the axiom of pairing implies that no set is an element of itself, and that there is no infinite sequence (''an'') such that ''ai+1'' is an element of ''ai'' 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 introduced by ; it was adopted in a formulation closer to the one found in contemporary te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiomatic Set Theory
Set theory is the branch of mathematical logic that studies 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 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 theory is commonly employed as a foundational s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-founded Relation
In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s R m'' (for instance, "''s'' is not smaller than ''m''") for any ''s'' ∈ ''S''. In other words, a relation is well founded if :(\forall S \subseteq X)\; \neq \emptyset \implies (\exists m \in S) (\forall s \in S) \lnot(s \mathrel m) Some authors include an extra condition that ''R'' is 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, which can be proved when there is no infinite sequence ''x''0, ''x''1, ''x''2, ... of elements of ''X'' such that ''x''''n''+1 ''R'' ''x''n for every natural number ''n''. In order theory, a partial order is called well-f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.Zermelo: ''Untersuchungen über die Grundlagen der Mengenlehre'', 1907, in: Mathematische Annalen 65 (1908), 261-281; Axiom des Unendlichen p. 266f. Formal statement In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: :\exists \mathbf \, ( \empty \in \mathbf \, \land \, \forall x \in \mathbf \, ( \, ( x \cup \ ) \in \mathbf ) ) . In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any ''x'' is a member of I, the set formed by taking the union of ''x'' with its singleton is also a member of I. Such a set is sometimes called an inductive set. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Set
In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions hold: * whenever x \in A, and y \in x, then y \in A. * whenever x \in A, and x is not an urelement, then x is a subset of A. Similarly, a class M is transitive if every element of M is a subset of M. Examples Using the definition of ordinal numbers suggested by John von Neumann, ordinal numbers are defined as hereditarily transitive sets: an ordinal number is a transitive set whose members are also transitive (and thus ordinals). The class of all ordinals is a transitive class. Any of the stages V_\alpha and L_\alpha leading to the construction of the von Neumann universe V and Gödel's constructible universe L are transitive sets. The universes V and L themselves are transitive classes. This is a complete list of all finite transitive sets with up to 20 brackets: * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zermelo Set Theory
Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It bears certain differences from its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering. The axioms of Zermelo set theory The axioms of Zermelo set theory are stated for objects, some of which (but not necessarily all) are sets, and the remaining objects are urelements and not sets. Zermelo's language implicitly includes a membership relation ∈, an equality relation = (if it is not included in the underlying logic), and a unary predicate saying whether an object is a set. Later versions of set theory often assume that all objects are sets so there are no urelements and there is no need for the unary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thoralf Skolem
Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. Life Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania (later renamed Oslo), passing the university entrance examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study mathematics, also taking courses in physics, chemistry, zoology and botany. In 1909, he began working as an assistant to the physicist Kristian Birkeland, known for bombarding magnetized spheres with electrons and obtaining aurora-like effects; thus Skolem's first publications were physics papers written jointly with Birkeland. In 1913, Skolem passed the state examinations with distinction, and completed a dissertation titled ''Investigations on the Algebra of Logic''. He also traveled with Birkeland to the Sudan to observe the zodiacal light. He spent the winter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest coverage of any mathematician of his time and was said to have been "the last representative of the great mathematicians who were equally at home in both pure and applied mathematics". He integrated pure and applied sciences. Von Neumann made major contributions to many fields, including mathematics ( foundations of mathematics, measure theory, functional analysis, ergodic theory, group theory, lattice theory, representation theory, operator algebras, matrix theory, geometry, and numerical analysis), physics ( quantum mechanics, hydrodynamics, ballistics, nuclear physics and quantum statistical mechanics), economics ( game theory and general equilibrium theory), computing ( Von Neumann architecture, linear programming, numeri ... [...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 from conta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Excluded Middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradiction, and the law of identity. However, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law (or principle) of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'': "no third ossibilityis given". It is a tautology. The principle should not be confused with the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable now, but provable in the futur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
