Difference between revisions of "Universal Set"

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== References ==
 
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=== 2019 ===
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* (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Universe_(mathematics) Retrieved:2019-11-10.
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** In [[mathematics]], and particularly in [[set theory]], [[category theory]], [[type theory]], and the [[foundations of mathematics]], a '''universe''' is a collection that contains all the entities one wishes to consider in a given situation. It is related to the concept of a [[domain of discourse]] in [[philosophy]]. <P> In set theory, universes are often [[class (set theory)|classes]] that contain (as [[element (set theory)|elements]]) all sets for which one hopes to [[Mathematical proof|prove]] a particular [[theorem]]. These classes can serve as [[Inner model|inner models]] for various axiomatic systems such as [[Zermelo–Fraenkel set theory|ZFC]] or [[Morse–Kelley set theory]]. Universes are of critical importance to formalizing concepts in [[category theory]] inside set-theoretical foundations. For instance, the [[List of mathematical jargon#canonical|canonical]] motivating example of a category is '''[[Category of sets|Set]]''', the category of all sets, which cannot be formalized in a set theory without some notion of a universe. <P> In type theory, a universe is a type whose elements are types.
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=== 2019 ===
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* (Wikipedia, 2019) &rArr; https://en.wikipedia.org/wiki/Universal_set Retrieved:2019-11-10.
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** In [[set theory]], a '''universal set''' is a set which contains all objects, including itself. <ref> Forster 1995 p. 1. </ref> In set theory as usually formulated, the conception of a universal set leads to [[Russell's paradox]] and is consequently not allowed. However, some non-standard variants of set theory include a universal set.

Revision as of 23:25, 10 November 2019

A Universe Set is a set that contains all Things, including itself.



References

2011

In set theory, a universal set is a set which contains all objects, including itself.[1] In set theory as usually formulated, the conception of a set of all sets leads to a paradox. The reason for this lies with Zermelo's axiom of separation: for any formula [math]\varphi(x)[/math] and set [math]A[/math], the set [math]\{x \in A \mid \varphi(x)\}[/math] which contains exactly those elements [math]x[/math] of [math]A[/math] that satisfy [math]\varphi[/math] exists. If the universal set [math]V[/math] existed and the axiom of separation applied to it, then Russell's paradox would arise from [math]\{x \in V\mid x\not\in x\}[/math]. More generally, for any set [math]A[/math] we can prove that [math]\{x \in A\mid x\not\in x\}[/math] is not an element of A.

A second issue is that the power set of the set of all sets would be a subset of the set of all sets, providing that both exist. This conflicts with Cantor's theorem that the power set of any set (whether infinite or not) always has strictly higher cardinality than the set itself.


2019

2019

  • (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Universal_set Retrieved:2019-11-10.
    • In set theory, a universal set is a set which contains all objects, including itself. [2] In set theory as usually formulated, the conception of a universal set leads to Russell's paradox and is consequently not allowed. However, some non-standard variants of set theory include a universal set.
  • Forster 1995 p. 1.
  • Forster 1995 p. 1.