Table of contents |
2 In ordinary mathematics 3 In set theory 4 In category theory |
There are several precise versions of this general idea. Perhaps the simplest is that any set can be a universe, so long as you are studying that particular set. So if you're studying the real numbers, then the real line R, which is the set of all real numbers, could be your universe. 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 A's circle. Strictly speaking, this is the relative complement U \\ A of A relative to U; but in a context where U is the universe, we can regard this as this as the absolute complement A^{C} of A. Similarly, we have a notion of the nullary intersection, that is the intersection of zero sets. Without a universe, the nullary intersection would be the set of absolutely everything, which is generally regarded as impossible; but with the universe in mind, we can treat the nullary intersection as the set of everything under consideration, which is simply U.
These conventions are quite useful in the algebraic approach to basic set theory, based on Boolean lattices. Except in some non-standard forms of axiomatic set theory (such as New Foundations, the class of all sets is not a Boolean lattice (it is only a relatively complemented lattice). In contrast, the class of all subsets of U, called the power set of U, is a Boolean lattice. The absolute complement described above the is complement operation in the Boolean lattice; and U, as the nullary intersection, serves as the top element (or nullary meet) in the Boolean lattice. Then De Morgan's laws, which deal with complements of meets and joins (which are unionss in set theory) apply, and apply even to the nullary meet and the nullary join (which is the empty set).
However, once you consider subsets of a given set X (in Cantor's case, X = R), you may become interested in sets 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. Of course, it doesn't stop there; you might next be interested in sets of sets of subsets of X, and so on. In another direction, you may become interested in the Cartesian product X × X, or in functions from X to itself. Then you might want functions on the Cartesian product, or from X to X × PX, and so on.
Thus even if your primary interest is X, you may well want your universe to be quite a bit larger than X. Following the above ideas, you may want the superstructure over X. This can be defined by structural recursion as follows:
Note that no matter what set X you start with, the empty set {} will belong to S_{1}. Recall that the empty set is the von Neumann ordinal [0]. Then {[0]}, the set whose only element is the empty set, will belong to S_{2}; this is the von Neumann ordinal [1]. Similarly, {[1]} will belong to S_{3}, 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 , which represents the ordered pair (x,y). Thus the superstructure will contain the various desired Cartesian products. Then the superstructure also contains functions and relationss, since these may be represented as subsets of Cartesian products. We also get ordered n-tuples, represented as functions whose domain is the von Neumann ordinal [n]. And so on.
So if you start with just X = {}, then you can build up a great deal of the sets needed for mathematics as the elements of the superstructure over {}. But all of the elements of S_{{}} will be finite sets! All of the natural numbers belong 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, we 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 we may have to start the process all over again and form S_{S{}}. However, to keep things simple, let's just take the set N of natural numbers as given and form S_{N}, 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) will belong to S_{N}. Even nonstandard analysis can be done in the superstructure over a nonstandard 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 PS is a Boolean lattice, what is relevant is that S 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 S; then PA is a subset of S (and in fact belongs to S).
We can give a precise meaning to the claim that S_{N} 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 Zermelo set theory proved insufficient for the further development of axiomatic set theory and other work in the foundations of mathematics, especially model theory. For a dramatic example, the description of the superstructure process above cannot itself be carried out in Zermelo set theory! The final step, forming S as a infinitary union, requires the axiom of replacement, which was added to Zermelo set theory in 1922 to form Zermelo-Fraenkel set theory, the set of axioms most widely accepted today. So while ordinary mathematics may be done in S_{N}, discussion of S_{N} goes beyond the "ordinary".