Universal Set

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

• AKA: Set Universe.
• Context:
• It is usually denoted by the capital letter, [math]\displaystyle{ U }[/math].
• Example(s):
  • The complement of an empty set.
  • …
• Counter-Example(s):
  • an Empty Set,
  • a Subset.
• See: Complement Set Operation, Grothendieck Universe, Domain of Discourse.

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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. 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

• (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.

2015

• (Tsokos & Wooten, 2015) & Chris P. Tsokos, Rebecca D. Wooten (2015). "The Joy of Finite Mathematics: The Language and Art of Math". Academic Press.
  • QUOTE: The universal set [math]\displaystyle{ U }[/math] is the largest set in given context that is; the universal set is the totality of the elements under consideration. Denoted by the capital letter, [math]\displaystyle{ U }[/math] is normally written with the upper bars, [math]\displaystyle{ U }[/math] or a tail, [math]\displaystyle{ U }[/math] to be it distinguishable from the union symbol, [math]\displaystyle{ \cup }[/math].

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