Universe (mathematics)

In mathematics, and particularly in set 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. 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 inside set-theoretical foundations. For instance, the canonical motivating example of a category is , 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. Domain of discourse Perhaps the simplest version is that any set can be a universe, so long as the object of study is confined to that particular set. If the object of study is formed by the real numbers, then the real line R, which is the real number set, could be the universe under consideration. Implicitly, this is the universe that Georg Cantor was using when he first developed modern naive set theory and cardinality in the 1870s and 1880s in applications to real analysis. The only sets that Cantor was originally interested in were subsets of R. This concept of a universe is reflected in the use of Venn diagrams. In a Venn diagram, the action traditionally takes place inside a large rectangle that represents the universe U. One generally says that sets are represented by circles; but these sets can only be subsets of U. The complement of a set A is then given by that portion of the rectangle outside of As circle. Strictly speaking, this is the relative complement U \ A of A relative to U; but in a context where U is the universe, it can be regarded as the absolute complement AC of A. Similarly, there is a notion of the nullary intersection, that is the intersection of zero sets (meaning no sets, not null sets).

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