Universe (mathematics)

Universes are of critical importance to formalizing concepts in category theory inside set-theoretical foundations.

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.

These conventions are quite useful in the algebraic approach to basic set theory, based on Boolean lattices.

Then De Morgan's laws, which deal with complements of meets and joins (which are unions in set theory) apply, and apply even to the nullary meet and the nullary join (which is the empty set).

This may be continued; the object of study may next consist of such sets of subsets of X, and so on, in which case the universe will be P(PX).

In another direction, the binary relations on X (subsets of the Cartesian product X × X) may be considered, or functions from X to itself, requiring universes like P(X × X) or XX.

Continuing this process, every natural number is represented in the superstructure by its von Neumann ordinal.

The process also gives ordered n-tuples, represented as functions whose domain is the von Neumann ordinal [n], and so on.

So if the starting point is just X = {}, a great deal of the sets needed for mathematics appear as elements of the superstructure over {}.

Speaking anachronistically, one could suggest that the 19th-century finitist Leopold Kronecker was working in this universe; he believed that each natural number existed but that the set N (a "completed infinity") did not.

However, to keep things simple, one can take the set N of natural numbers as given and form SN, the superstructure over N. This is often considered the universe of ordinary mathematics.

It is possible to give a precise meaning to the claim that SN is the universe of ordinary mathematics; it is a model of Zermelo set theory, the axiomatic set theory originally developed by Ernst Zermelo in 1908.

Zermelo set theory was successful precisely because it was capable of axiomatising "ordinary" mathematics, fulfilling the programme begun by Cantor over 30 years earlier.

But if high-powered set theory is brought in, the superstructure process above reveals itself to be merely the beginning of a transfinite recursion.

But what used to be called "superstructure" is now just the next item on the list: Vω, where ω is the first infinite ordinal number.

The point of this axiom is that any set one encounters is then U-small for some U, so any argument done in a general Grothendieck universe can be applied.

It gives rise to a “reflection principle which roughly speaking says whatever we are used to doing with types can be done inside a universe” (Martin-Löf 1975, 83).