# Difference between revisions of "Universal Set"

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− | A [[ | + | A [[Universal Set]] is a [[set]] that contains all [[objects]]s, including itself. |

− | * <B>AKA:</B> [[Set Universe | + | * <B>AKA:</B> [[Set Universe]]. |

* <B>See:</B> [[Complement Set Operation]], | * <B>See:</B> [[Complement Set Operation]], | ||

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== References == | == References == | ||

− | + | === 2019a === | |

− | === | + | * (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. <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. |

− | + | === 2019b === | |

− | In [[set theory]], a '''universal set | + | * (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Universe_(mathematics) Retrieved:2019-11-10. |

− | + | ** 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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## Revision as of 23:27, 10 November 2019

A Universal Set is a set that contains all objectss, including itself.

**AKA:**Set Universe.**See:**Complement Set Operation,

## References

### 2019a

- (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.^{[1]}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.

- In set theory, a

### 2019b

- (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Universe_(mathematics) Retrieved:2019-11-10.
- 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.In set theory, universes are often classes that contain (as elements) all sets for which one hopes to prove a particular theorem. These classes can serve as inner models for various axiomatic systems such as ZFC or Morse–Kelley set theory. Universes are of critical importance to formalizing concepts in category theory inside set-theoretical foundations. For instance, the canonical motivating example of a category is

**Set**, the category of all sets, which cannot be formalized in a set theory without some notion of a universe.In type theory, a universe is a type whose elements are types.

- In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a

- ↑ Forster 1995 p. 1.