Universe (mathematics)

Lore & Background

The simplest version of a universe is any set that confines the object of study; for example, the real line R served as the universe when Georg Cantor developed naive set theory and cardinality in the 1870s and 1880s. In Venn diagrams, the universe is represented by a large rectangle, with sets as subsets of that universe; the complement of a set is then the relative complement U \ A, and the nullary intersection is the universe itself. These conventions are useful in the algebraic approach to set theory based on Boolean lattices.

In ordinary mathematics, the universe may need to be a superstructure over a starting set X, defined recursively: S0X = X, S1X = X ∪ PX, and so on, with the superstructure SX being the union over all stages. For example, starting from the empty set yields the hereditarily finite sets, suitable for finitist mathematics. Taking N as given and forming SN is often considered the universe of ordinary mathematics, containing constructions like Dedekind cuts for real numbers.

In set theory, SN is a model of Zermelo set theory, which axiomatized ordinary mathematics. However, the superstructure process itself cannot be carried out in Zermelo set theory; it requires the axiom of replacement, added in 1922 to form Zermelo–Fraenkel set theory.

Reader's Guide

The concept of a universe is foundational to modern mathematics, providing a framework within which sets, relations, and functions can be studied without paradox. In set theory, universes allow the formalization of categories like Set, the category of all sets, which cannot be defined without a universe. The superstructure construction over a set X, such as N, yields a universe containing all ordinary mathematical objects, including real numbers and functions. This approach underpins Zermelo set theory, which successfully axiomatized ordinary mathematics, but the construction of the superstructure itself requires the axiom of replacement, leading to Zermelo–Fraenkel set theory. The notion of a universe also appears in type theory, where a universe is a type whose elements are types. Despite its utility, the concept is not without limitations: the class of all sets is not a Boolean lattice in standard axiomatic set theory, and the superstructure over the empty set does not contain the power set of N, necessitating larger universes for advanced mathematics.

Did You Know?

• In Venn diagrams, the universe is represented by a large rectangle, and the complement of a set is the relative complement U \ A.
• The superstructure over the empty set consists of all hereditarily finite sets, which can be considered the universe of finitist mathematics.
• The superstructure process cannot be carried out in Zermelo set theory; it requires the axiom of replacement, added in 1922 to form Zermelo–Fraenkel set theory.
• In type theory, a universe is a type whose elements are types.