Conglomerate (set theory)

In mathematics, a conglomerate is a collection of classes, just as a class is a collection of sets.

Conglomerates are used to provide a set-theoretic foundation for category theory. Some naturally occuring entities in category theory, such as the category of all categories, cannot be formed without leading to paradoxes like Russell's paradox. For this reason, a new concept is introduced: a quasi-category. A quasi-category is like a category except that its objects and morphisms form conglomerates instead of classes.

Like classes, conglomerates cannot be defined in some set theories, such as Zermelo–Fraenkel set theory (ZFC), and so one needs to extend a set theory to allow them. As Gödel-Bernays set theory extends ZFC to allow classes, there are extensions that allow conglomerates.

Conglomerate theory

There are different ways to introduce conglomerates. For category theory, the details of the definition are not important but the following properties are required:

• Every class is a conglomerate.
• A collection of all classes that satisfy a particular property forms a conglomerate.
  • In particular, it means that all classes form a conglomerate.
• Conglomerates are closed under the usual set-theoretic operations such as pairing, unions, product and power.

Also, each sub-conglomerate of a class should be a class. A theory that satisfies these conditions is called a conglomerate theory and, if conglomerates satisfy all ZFC axioms, then it is called a strong conglomerate theory. The strong conglomerate theory is consistent if, and only if, the existence of strongly inaccessible cardinals is undecidable (i.e. it cannot be proved nor disproved) in ZFC (assuming ZFC is consistent).

Conglomerates and Grothendieck universes

One way to define conglomerates is to extend ZFC by postulating the existence of a Grothendieck universe (which, by the way, implies the existence of strongly inaccessible cardinals). Suppose one such universe ${\mathcal {U}}$ is fixed. The following conglomerate conversion for ${\mathcal {U}}$ is used:

• The elements of ${\mathcal {U}}$ are called sets (or small set).
• The subsets of ${\mathcal {U}}$ are called classes.
• And all original sets of our extended ZFC are now called conglomerates.

Thus the original ZFC becomes a subset of ZFC with a Grothendieck universe. According to this notation, all sets are classes and all classes are conglomerates. There is a class of all sets (${\mathcal {U}}$) which is not a set, and the conglomerate of all classes (${\mathcal {P}}({\mathcal {U}})$) is not a class. Moreover, for any property $P$ there is a conglomerate of all classes that satisfy this property $P$.

Axiomatic systems with conglomerates

Another way to define conglomerates is to extend a set theory by adding special axioms that introduce classes and conglomerates. An example of such a system is ACG – a non-conservative extension of Gödel-Bernays set theory. This system is formulated as many-sorted first-order logic for different sorts for sets, classes and conglomerates. It has the axioms of Gödel-Bernays set theory with some additional axioms. In this system, one can prove that every set is a class and every class is a conglomerate. Every set belongs to some class and every class belongs to some conglomerate. However the collection of all conglomerates is not a conglomerate. Conglomerates in this system are closed under pairing, unions, product and power.

ACG is a non-conservative extension of Quine-Morse set theory and ZF# (Zermelo–Fraenkel set theory with an axiom of existence of inaccessible cardinals). Moreover, it can prove the consistency of both of these theories.

Beyond conglomerates

It is also possible to introduce higher level of entities in addition to sets, classes, and conglomerates. Then one can define for example quasi-quasicategory of quasicategories. However this is rarely needed.

In Martin-Löf type theory there is a hierarchy of universes ${\mathcal {U}}_{i}$, for any natural number $i$: ${\mathcal {U}}_{0}$ is a type of "small types", i.e. types that can be formed without referring to a type of types. ${\mathcal {U}}_{1}$ contains ${\mathcal {U}}_{0}$ and all types that be formed using small types and ${\mathcal {U}}_{0}$. For example a type of groups or functions ${\mathcal {U}}_{0}\rightarrow {\mathcal {U}}_{0}$. In general ${\mathcal {U}}_{i+1}$ contains ${\mathcal {U}}_{i}$. Universe types are a tricky feature of type theories. Martin-Löf's original type theory had to be changed to account for Girard's paradox which is the type-theoretic analogue of Russell's paradox.

References

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