Conglomerate (mathematics)


In mathematics, in the framework of one-universe foundation for category theory, the term "conglomerate" is applied to arbitrary sets as a contraposition to the distinguished sets that are elements of a Grothendieck universe.

Definition

The most popular axiomatic set theories, Zermelo–Fraenkel set theory, von Neumann–Bernays–Gödel set theory, and Morse–Kelley set theory, admit non-conservative extensions that arise after adding a supplementary axiom of existence of a Grothendieck universe. An example of such an extension is the Tarski–Grothendieck set theory, where an infinite hierarchy of Grothendieck universes is postulated.
The concept of conglomerate was created to deal with "collections" of classes, which is desirable in category theory so that each class can be considered as an element of a "more general collection", a conglomerate. Technically this is organized by changes in terminology: when a Grothendieck universe is added to the chosen axiomatic set theory it is considered convenient
As a result, in this terminology, each set is a class, and each class is a conglomerate.

Corollaries

Formally this construction describes a model of the initial axiomatic set theory in the extension of this theory with as the universe.
If the initial axiomatic set theory admits the idea of proper class, then these objects are discarded from the consideration in the new theory. However, this in some sense does not lead to a loss of information about objects of the old theory since its representation as a model in the new theory means that what can be proved in NBG/MK about its usual objects called classes can be proved as well in "NBG/MK+Grothendieck universe" about its classes. At the same time, the new theory is not equivalent to the initial one, since some extra propositions about classes can be proved in "NBG/MK+Grothendieck universe" but not in NBG/MK.

Terminology

The change in terminology is sometimes called "conglomerate convention".
The first step, made by Mac Lane, is to apply the term "class" only to subsets of Mac Lane does not redefine existing set-theoretic terms; rather, he works in a set theory without classes, calls members of "small sets", and states that the small sets and the classes satisfy the axioms of NBG. He does not need "conglomerates", since sets need not be small.
The term "conglomerate" lurks in reviews of 1970th and 1980th on Mathematical Reviews without definition, explanation or reference, and sometimes in papers.
While the conglomerate convention is in force, it must be used exclusively in order to avoid ambiguity; that is, conglomerates should not be called “sets” in the usual fashion of ZFC.