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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