NationMaster - Encyclopedia: List of axioms

This is a list of axioms as that term is understood in mathematics, by Wikipedia page. In epistemology, the word axiom is understood differently; see axiom and self-evidence. Individual axioms are almost always part of a larger axiomatic system. An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths. ... Euclid, detail from The School of Athens by Raphael. ... Epistemology is an analytic branch of philosophy which studies the nature, origin, and scope of knowledge. ... An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths. ... In epistemology, a self-evident proposition is one that can be understood only by one who knows that it is true. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ...

Zermelo-Frankel axioms

These are the de facto standard axioms for contemporary mathematics or set theory. They can be easily adapted to analogous theories, such as mereology. The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... De facto is a Latin expression that means in fact or in practice. It is commonly used as opposed to de jure (meaning by law) when referring to matters of law or governance or technique (such as standards), that are found in the common experience as created or developed without... Euclid, detail from The School of Athens by Raphael. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Mereology is a collection of axiomatic formal systems dealing with parts and their respective wholes. ...

See also Zermelo set theory. In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of empty set is one of the axioms of Zermelo_Fraenkel set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo-Fraenkel set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of union is one of the axioms of Zermelo_Fraenkel set theory, stating that, for any two sets, there is a set that contains exactly the elements of both. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory. ... In mathematics, the axiom of power set is one of the Zermelo-Fraenkel axioms of axiomatic set theory. ... The axiom of regularity (also known as the axiom of foundation) is one of the axioms of Zermelo-Fraenkel set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, or axiom schema of separation, or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, or axiom schema of separation, or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory. ... Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. ...

Axiom of choice

With the Zermelo-Frankel axioms above, this makes up the system ZFC in which most mathematics is potentially formalisable In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

Equivalents of AC

The Hausdorff maximality theorem, formulated and proved by Felix Hausdorff in 1914, is an alternate formulation of Zorns lemma and therefore also equivalent to the axiom of choice. ... Sometimes the phrase well-ordering principle (or the axiom of choice) is taken to be synonymous with well-ordering theorem. On other occasions the phrase is taken to mean the proposition that the set of natural numbers {1, 2, 3, ....} is well-ordered, i. ... Zorns lemma, also known as the Kuratowski-Zorn lemma, is a theorem of set theory that states: Every non-empty partially ordered set in which every chain (i. ...

Weaker than AC

The axiom of countable choice or axiom of denumerable choice is an axiom of set theory similar to the axiom of choice. ... In mathematics, the axiom of dependent choice is a weak form of the axiom of choice which is still sufficient to develop most of real analysis. ... In mathematics, a number of prime ideal theorems for guaranteeing the existence of certain subsets of an abstract algebra can be stated. ... In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that if is a subset of , where and are Polish spaces, then there is a subset of that is a partial function from to , and whose domain equals Such a function is called a...

Alternates incompatible with AC

In mathematics, the axiom of real determinacy (abbreviated as ADR) is an axiom in set theory. ...

Other axioms of mathematical logic

Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... In foundations of mathematics, Von Neumann-Bernays-GÃ¶del set theory (NBG) is an axiom system for set theory designed to yield the same results as Zermelo-Fraenkel set theory, together with the axiom of choice (ZFC), but with only a finite number of axioms, that is without axiom schemas. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... Freilings axiom of symmetry (AX) is a set-theoretic axiom proposed by Chris Freiling. ... In mathematics, the axiom of determinacy (abbreviated as AD) is an axiom in set theory. ... In mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only to projective sets. ... In axiomatic set theory, Martins axiom is a statement which is independent of the usual axioms of ZFC Set Theory. ... The axiom of constructibility is a possible axiom for set theory in mathematics. ... In set theory, a rank-into-rank is a large cardinal λ satisfying one of the following four axioms (commonly known as rank-into-rank embeddings, given in order of increasing consistency strength): There is a nontrivial elementary embedding of Vλ into itself. ... The Kripke-Platek axioms of set theory (KP) are a system of axioms of axiomatic set theory. ...

Geometry

Table of Geometry, from the 1728 Cyclopaedia. ... a and b are parallel, the transversal t produces congruent angles. ... In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoffs axioms. ... Hilberts axioms are a set of 20 assumptions (originally 21), David Hilbert proposed in 1899 as the foundation for a modern treatment of Euclidean geometry. ... Tarskis axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called elementary, that is formulable in first order logic with identity, and requiring no set theory. ...

Other axioms

Categories: Mathematical lists | Mathematical axioms This article or section contains information that has not been verified and thus might not be reliable. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In mathematics, the fundamental axiom of analysis (or simply the fundamental axiom) states: Every non-decreasing sequence of real numbers which is bounded above tends to a limit. ... Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... In mathematics, the gluing axiom is introduced to define what a sheaf F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor F: Open(X) → C to a category C which initially one takes to be the category of... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... The Haag-Kastler axiomatic framework for quantum field theory is an application to local quantum physics of C-star algebra theory. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... Huzitas axioms are a set of rules related to the mathematical principles of paper folding. ... The traditional crane and papers of the same size used to fold it A paper pegasus designed by F. Kawahata Origami (æŠ˜ã‚Šç´™ or æŠ˜ç´™ origami paper folding) is the art of paper folding. ... In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... The probability P of some event E, denoted , is defined with respect to a universe, or sample space , of all possible elementary events in such a way that P must satisfy the Kolmogorov axioms. ... In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In physics the Wightman axioms are an attempt of mathematically stringent, axiomatic formulation of quantum field theory. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

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