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Axiomatic Reasoning
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system (usually formulated within model theory) that describes a set of sentences that is closed under logical implication. A formal proof is a complete rendition of a mathematical proof within a formal system. Properties An axiomatic system is said to be ''consistent'' if it lacks contradiction. That is, it is impossible to derive both a statement and its negation from the system's axioms. Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (principle of explosion). In an axiomatic system, ... [...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 points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line (geometry)
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segment in everyday life, which has two points to denote its ends. Lines can be referred by two points that lay on it (e.g., \overleftrightarrow) or by a single letter (e.g., \ell). Euclid described a line as "breadthless length" which "lies evenly with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry). In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality Of The Continuum
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \mathbb R, . The real numbers \mathbb R are more numerous than the natural numbers \mathbb N. Moreover, \mathbb R has the same number of elements as the power set of \mathbb N. Symbolically, if the cardinality of \mathbb N is denoted as \aleph_0, the cardinality of the continuum is This was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his diagonal argument in 1891. Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them. Between any two real numbers ''a'' \mathfrak c . Alternative explanation for 𝔠 = 2ℵ0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted , A, , with a vertical bar on each side; this is the same notation as absolute value, and the meaning depends on context. The cardinality of a set A may alternatively be denoted by n(A), , \operatorname(A), or \#A. History A crude sense of cardinality, an awareness that groups of things or events compare with other grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Properties The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC), but only by showing that it follows from the existence of the natural numbers. A set is infinite if and only if for every natural number, the set has a subset whose cardinality is that natural number. If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset. If a set of sets is infinite or contains an infinite element, then its union is infinite. The power set of an infinite set is infinite. Any superset of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Schema
In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which may or may not be required to satisfy certain conditions. Often, such conditions require that certain variables be free, or that certain variables not appear in the subformula or term. Finite axiomatization Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms. This set can usually be defined recursively. A theory that can be axiomatized without schemata is said to be finitely axiomatized. Theories that can be finitely axiomatized are seen as a bit more metamathematically elegant, even if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countably Infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (its number of elements) is not greater than that of the natural numbers. A countable set that is not finite is said countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defined here are quite com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists''"'' is a quantifier, while ''x'' is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of ax ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semantics
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ..., linguistics and computer science. History In English, the study of meaning in language has been known by many names that involve the Ancient Greek word (''sema'', "sign, mark, token"). In 1690, a Greek rendering of the term ''semiotics'', the interpretation of signs and symbols, finds an early allusion in John Locke's ''An Essay Concerning Human Understanding'': The third Branch may be called [''simeiotikí'', "semiotics"], or the Doctrine of Signs, the most usual whereof being words, it is aptly enough ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consistency Proof
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term ''satisfiable'' is used instead. The syntactic definition states a theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences of T. Let A be a set of closed sentences (informally "axioms") and \langle A\rangle the set of closed sentences provable from A under some (specified, possibly implicitly) formal deductive system. The set of axioms A is consistent when \varphi, \lnot \varphi \in \langle A \rangle for no formula \varphi. If there ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
