Difference between revisions of "Universal Set"

From GM-RKB
Jump to: navigation, search
(2019b)
Line 21: Line 21:
 
* (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Universe_(mathematics) Retrieved:2019-11-10.
 
* (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.
 
** 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.
 +
=== 2015 ===
 +
* (Tsokos & Wooten, 2015) & Chris P. Tsokos, Rebecca D. Wooten (2015). [https://books.google.ca/books?id=zu7HBQAAQBAJ "The Joy of Finite Mathematics: The Language and Art of Math"]. Academic Press.
 +
** QUOTE: The [[universal set]] <math>U</math> is the largest [[set]] in given context that is; the [[universal set]] is the totality of the [[element]]s under consideration. Denoted by the capital letter, <math>U</math> is normally written with the upper bars, <math>U</math>  or a tail, <math>U</math>  to be it distinguishable from the [[union symbol]], <math>\cup</math>.
 
----
 
----
  
 
__NOTOC__
 
__NOTOC__
 
[[Category:Concept]]
 
[[Category:Concept]]

Revision as of 23:48, 10 November 2019

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



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.

2019b

2015


  1. Forster 1995 p. 1.