Universe (mathematics) - in Ordinary Mathematics

In Ordinary Mathematics

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:

• Let S₀X be X itself.
• Let S₁X be the union of X and PX.
• Let S₂X be the union of S₁X and P(S₁X).
• In general, let S_n+1X be the union of S_nX and P(S_nX).

Then the superstructure over X, written SX, is the union of S₀X, S₁X, S₂X, and so on; or

Note that no matter what set X is the starting point, the empty set {} will belong to S₁X. The empty set is the von Neumann ordinal . Then {}, the set whose only element is the empty set, will belong to S₂X; this is the von Neumann ordinal . Similarly, {} will belong to S₃X, and thus so will {,}, as the union of {} and {}; this is the von Neumann ordinal . 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 . 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 finite sets! 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 X 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.

One should note 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.