Universe (mathematics)
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.
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.
However, once subsets of a given set X (in Cantor's case, X = R) are considered, the universe may need to be a set of subsets of X. (For example, a topology on X is a set of subsets of X.) The various sets of subsets of X will not themselves be subsets of X but will instead be subsets of PX, the power set of X. 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.
Thus, even if the primary interest is X, the universe may need to be considerably larger than X. Following the above ideas, one may want the superstructure over X as the universe. This can be defined by structural recursion as follows:
Then the superstructure over X, written SX, is the union of S0X, S1X, S2X, and so on; or
No matter what set X is the starting point, the empty set {} will belong to S1X. The empty set is the von Neumann ordinal [0].
Then {[0]}, the set whose only element is the empty set, will belong to S2X; this is the von Neumann ordinal [1]. Similarly, {[1]} will belong to S3X, and thus so will {[0],[1]}, as the union of {[0]} and {[1]}; this is the von Neumann ordinal [2]. Continuing this process, every natural number is represented in the superstructure by its von Neumann ordinal. Next, if x and y belong to the superstructure, then so does {{x},{x,y}}, which represents the ordered pair (x,y). Thus the superstructure will contain the various desired Cartesian products. Then the superstructure also contains functions and relations, since these may be represented as subsets of Cartesian products. 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 {}. But each of the elements of S{} will be a finite set. Each of the natural numbers belongs to it, but the set N of all natural numbers does not (although it is a subset of S{}). In fact, the superstructure over {} consists of all of the hereditarily finite sets. As such, it can be considered the universe of finitist mathematics. 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, S{} is unsatisfactory for ordinary mathematicians (who are not finitists), because even though N may be available as a subset of S{}, still the power set of N is not. In particular, arbitrary sets of real numbers are not available. So it may be necessary to start the process all over again and form S(S{}). 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. The idea is that all of the mathematics that is ordinarily studied refers to elements of this universe. For example, any of the usual constructions of the real numbers (say by Dedekind cuts) belongs to SN. Even non-standard analysis can be done in the superstructure over a non-standard model of the natural numbers.
There is a slight shift in philosophy from the previous section, where the universe was any set U of interest. There, the sets being studied were subsets of the universe; now, they are members of the universe. Thus although P(SX) is a Boolean lattice, what is relevant is that SX itself is not. Consequently, it is rare to apply the notions of Boolean lattices and Venn diagrams directly to the superstructure universe as they were to the power-set universes of the previous section. Instead, one can work with the individual Boolean lattices PA, where A is any relevant set belonging to SX; then PA is a subset of SX (and in fact belongs to SX). In Cantor's case X = R in particular, arbitrary sets of real numbers are not available, so there it may indeed be necessary to start the process all over again.
In predicate calculus[edit]
In an interpretation of first-order logic, the universe (or domain of discourse) is the set of individuals (individual constants) over which the quantifiers range. A proposition such as ∀x (x2 ≠ 2) is ambiguous, if no domain of discourse has been identified. In one interpretation, the domain of discourse could be the set of real numbers; in another interpretation, it could be the set of natural numbers. If the domain of discourse is the set of real numbers, the proposition is false, with x = √2 as counterexample; if the domain is the set of naturals, the proposition is true, since 2 is not the square of any natural number.