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...out to be a tricky matter. However, some unproblematic examples from naïve set theory will make the concept clearer. These examples will be used throughout this == History of set theory ==

...e notion of comparison between [[number]]s and magnitudes, or [[inclusion (set theory)|inclusion]] between sets or [[algebraic structure]]s. ...maximal within the family of chains ordered by set-theoretic [[inclusion (set theory)|inclusion]]).

...lved.) The very existence of various sets introduced below is addressed by set theory, for example by the [[Zermelo-Fraenkel axioms]].<ref name=Jech/> See {{cite book |title=Naive set theory |author=Paul Richard Halmos |chapter=Section 9: Families |url=http://books.

...mes '''Cantor-Schröder-Bernstein theorem''') is a fundamental theorem of [[set theory]]. * '''1895''' [[Georg Cantor]] states the theorem in his first paper on set theory and transfinite numbers (as an easy consequence of the linear order of card

...mes '''Cantor-Schröder-Bernstein theorem''') is a fundamental theorem of [[set theory]]. * '''1895''' [[Georg Cantor]] states the theorem in his first paper on set theory and transfinite numbers (as an easy consequence of the linear order of card

...out to be a tricky matter. However, some unproblematic examples from naïve set theory will make the concept clearer. These examples will be used throughout this == History of set theory ==

who showed that &ndash; in set theory including the [[axiom of choice]] &ndash; ...othesis is independent of the usual [[axiomatic set theory|(ZFC) axioms of set theory]].

...e subjects. For example, arithmetic has the product of a pair of numbers, set theory has the Cartesian product of a pair of sets and logic has the conjunction o

...b> is a subset of ''E''_''n''+1 for all ''n'', then the [[Union (set theory)|union]] of the sets ''E''_''n'' is measurable, and ...a subset of ''E''_''n'' for all ''n'', then the [[Intersection (set theory)|intersection]] of the sets ''E''_''n'' is measurable; furthermor

...s [[nitrogen]]", and unlike other explanations - realist or nominalist - [[set theory]] provides a mature understanding of classes including identity conditions.

...ann Benedict Listing]]. Modern topology depends strongly on the ideas of [[set theory]], developed by [[Georg Cantor]] in the later part of the 19th century. [[H

or, in set theory, as a specific set that serves as a concrete object (model) In modern mathematics, in particular because of set theory and

* Zimmermann H., ''Fuzzy Set Theory and its Applications'' (2001), ISBN 0-7923-7435-5.

In the set theory, infinity appears directly; for instance,

the sum being taken on ''E''₁ of the ''d'' points on the [[fibre (set theory)|fibre]] over ''Q''. This is indeed an isogeny, and the [[function composi

...alogy to the [[Schröder-Bernstein theorem|theorem]] of the same name (from set theory).

...alogy to the [[Schröder-Bernstein theorem|theorem]] of the same name (from set theory).

...defined object that underlies some [[Set_theory#The_paradoxes|paradoxes in set theory]]. The idea of a universe ''U'' need not be paradoxical, however, if one co

...Octonion]]s were discovered in 1843. [[Georg Cantor]], through its naive [[set theory]], formally defined the notion of [[infinity]] in 1895. [[Kurt Hensel]] fir

...>, we define the ''closed sets'' of <math>X</math> to be the [[complement (set theory)|complement]]s (in <math>X</math>) of the open sets; the closed sets of <ma

==Functions in set theory== In [[set theory]], functions are regarded as a special class of [[relation (mathematics)|re

* Zimmermann H., ''Fuzzy Set Theory and its Applications'' (2001), ISBN 0-7923-7435-5.

...for α. For example, let ʘ be the minimum, ''T'' be a system of axioms for set theory such that the choice axiom ''CA'' does not depend on ''T''. Then we can co Then, despite the fact that no vague predicate is considered in set theory, in the metalanguage we can consider a vague meta-predicate as "is acceptab

...p]]s that are the group of integers [[modular arithmetic|modulo]] ''n''. [[Set theory]] is a branch of [[logic]] and not technically a branch of algebra.

...gical independence|independent]] from the [[axiomatic set theory|axioms of set theory]].

...one may use untyped variables and express everything in the framework of [[set theory]] by means of guarded quantification. In this case maps and sets are reduce ...ld have to pay me money to make me twist my brain around HOL and its typed set theory.

...mportant work (equally significant, but less well known) was his work in [[set theory]], where he proved that [[Georg Cantor]]'s puzzling [[Continuum Hypothesis]

...n of mathematics]]. Also [[category theory]] pretends to the throne of the set theory.<ref>{{harvnb|Lawvere|Rosebrugh|2003}}.</ref> ...ed in mathematics two centuries later, especially, as a formal language of set theory. In the form introduced by Bourbaki this language contains four logical sig

One of Russell's primary contributions in philosophy, mathematics and [[set theory]] is [[Russell's Paradox]], which he discovered in 1901. The paradox is tha

...ncient to modern times, Volume 3 |chapter=Chapter 51: §2: The paradoxes of set theory |url=http://books.google.com/books?id=8YaBuGcmLb0C&pg=PA1183 |author=Morris ...ncient to modern times, Volume 3 |chapter=Chapter 51: §2: The paradoxes of set theory |url=http://books.google.com/books?id=8YaBuGcmLb0C&pg=PA1183 |author=Morris

...t every mathematical statement or proof could be cast into formulas within set theory. ...eart of mathematics to other fields: to [[Mathematical logic|logic]], to [[set theory]] ([[Foundations of mathematics|foundations]]), to the empirical mathematic

Hash tables are often used to implement dictionaries or [[set theory|sets]]. Internet [[router|routers]] usually use a hash table to correlate

...allowed the formalisation of mathematics, and drove the investigation of [[set theory]], allowed the development of [[Alfred Tarski]]'s approach to [[model theor ...tical logic, they have been but two of the four pillars of the subject. [[Set theory]] originated in the study of the infinite by [[Georg Cantor]], and it has b

...at the [[continuum hypothesis]] is consistent with the axioms of classical set theory. He was interested in the mathematical aspects of the [[theory of relativit

...ch class set'' is a collection of three or more pitches, and ''pitch class set theory'' studies characteristics of differently chosen sets. The idea is to genera ...e functionally the same, one octave apart). Among the conventions of music set theory are:<ref name=Mayfield/>

...y other forms. [[Musical set theory]] is the application of mathematical [[set theory]] to music, first applied to [[atonal music]]. [[Speculative music theory]]

...tribute tables of both inputs into a single new output. An [[intersection (set theory)|intersect]] overlay defines the area where both inputs overlap and retains

::: —Alec Rogers: ''Cognitive Set Theory, p. 85'' {{cite book |author=Alec Rogers |title=Cognitive Set Theory |url=http://books.google.com/books?id=qkEqfI3H4zkC&pg=PA85 |pages=p. 85 |ch

...universes ultimately contains the other."</font> –Alec Rogers: ''Cognitive Set Theory'', p.85<ref name=Rogers/> {{cite book |author=Alec Rogers |title=Cognitive Set Theory |url=http://books.google.com/books?id=qkEqfI3H4zkC&pg=PA85 |page=85 |chapte