Universe (mathematics)

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In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use.

Contents

In a specific context

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 AC of A. Similarly, we have a notion of the nullary intersection, that is the intersection of zero sets (meaning no sets, not null 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 is the 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 unions in set theory) apply, and apply even to the nullary meet and the nullary join (which is the empty set).

In ordinary mathematics

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:

  • 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 Sn+1X be the union of SnX and P(SnX).

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

<math> \mathbf{S}X := \bigcup_{i=0}^{\infty} \mathbf{S}_{i}X \mbox{.} \! <math>

Note that no matter what set X you start with, the empty set {} will belong to S1X. 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 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. 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 each of the elements of S{} will be finite sets! Each 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 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) will belong to SN. 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 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 set theory

We can 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 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 SN, discussion of SN goes beyond the "ordinary", into metamathematics.

But if we are going to bring in high-powered set theory, then we realise that the superstructure process above is merely the beginning of a transfinite recursion. Let us go back to X = {}, the empty set, and introduce the (standard) notation Vi for Si{}. Then we have V0 = {}, V1 = P{}, and so on as before. But what used to be called "superstructure" is now just the next item on our list: Vω, where ω is the first infinite ordinal number. If you're familiar with the ordinal numbers, then you know what happens next:

<math> V_{i} := \bigcup_{j

defines Vi for any ordinal number i. The union of all of the Vi is the Von Neumann universe V:

<math> V := \bigcup_{i} V_{i} \! <math>.

Note that every individual Vi is a set, but their union V is a proper class. The axiom of foundation, which was added to ZF set theory at around the same time as the axiom of replacement, says that every set belongs to V.

Kurt Gödel's constructible universe L and the axiom of constructibility
Inaccessible cardinals yield models of ZF and sometimes additional axioms

In category theory

There is another approach to universes which is historically connected with category theory. This is the idea of a Grothendieck universe. Conceptually, a Grothendieck universe is a set inside which all the usual operations of set theory can be performed. For example, the union of any two sets in a Grothendieck universe U is still in U. Similarly, intersections, unordered pairs, power sets, and so on are also in U. This is similar to the idea of a superstructure above. The advantage of a Grothendieck universe is that it is actually a set, and never a proper class; the disadvantage is that if one tries hard enough, one can leave a Grothendieck universe.

The most common use of a Grothendieck universe U is to take U as a replacement for the category of all sets. One says that a set S is U-small if SU, and U-large otherwise. The category U-Set of all U-small sets has as objects all U-small sets and as morphisms all functions between these sets. Both the object set and the morphism set are sets, so it becomes possible to discuss the category of "all" sets without invoking proper classes. Then we can define other categories in terms of our new category. For example, the category of all U-small categories is the category of all categories whose object set and whose morphism set are in U. Then the usual arguments of set theory are applicable to the category of all categories, and one does not have to worry about accidentally talking about proper classes. Because Grothendieck universes are extremely large, this suffices in almost all applications.

Often when working with Grothendieck universes, there is an axiom hiding in the background: "For all sets x, there exists a universe U such that xU." 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. This axiom is closely related to the existence of strongly inaccessible cardinals.

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