# Why don't we use conglomerates in abstract algebra? [closed]

Are there rings that are conglomerate "cardinality"? What structural properties do we lose passing from proper classes?

• By "conglomerates" do you mean proper classes? Sep 19 at 17:27
• @MichaelHardy No; I mean the next size up.
– anon
Sep 19 at 17:29
• You mean next after proper classes? Sep 19 at 17:29
• @anon can you think of any interesting such huge structure that justifies developing the general theory of such monsters? Not everything that can be named deserves to be studied. Sep 19 at 17:41
• Well, why was the definition of conglomerate introduced in the first place, and by whom? Sep 19 at 18:05

## 2 Answers

Assuming we're working in some appropriate theory capable of handling conglomerates (also called "hyperclasses" or "$$2$$-classes" in other sources) in a reasonable way, then yes, there are indeed conglomerate-sized "rings" (I'll drop the size hypothesis from the definition of a ring for simplicity going forwards).

There's nothing mysterious about this; for example, we can form the conglomerate-sized polynomial ring $$\mathbb{Z}[\mathfrak{X}]$$ given a conglomerate $$\mathfrak{X}$$ of (things we choose to construe as) indeterminates. Indeed, most of the time nothing very surprising happens when we consider "ultra-big" structures (although there are occasional exceptions). Offhand I don't know of any particularly interesting property which holds for class sized rings but not conglomerate sized rings.

• The surreal numbers are a really nice field and its a proper classes.
– anon
Sep 19 at 21:45
• @anon Sure, but note that they have "small" analogues: given any reasonably-closed transitive set $T$ we can look at the part of the surreal field living in $T$ and this will basically have the same relation to fields in $T$ that the whole surreal numbers have to set-sized fields. Sep 19 at 21:47

It is perfectly reasonable to model the whole "set,class,conglomerate,..." progression using the axiomatic system given by ZFC together with the axiom that there exists a proper class of strongly inaccessible cardinals. In this framework a "set" is just a set of cardinality less than the first inaccessible, a "class" is a set smaller than the second inaccessible, a "conglomerate", the third inaccessible, and so on. In this framework, often referred to as the axiom system of "Grothendieck universes", it's clear that there's a "homogeneity" that precludes any serious distinctions between results about class-sized and conglomerate-sized algebraic objects, as in the end these are all just really big uncountable sets. While other axiom systems like NBG do allow for a difference between sets and classes, to get anything more precise, you're going to have to specify an axiom system. In any case, the Grothendieck universes formalization should serve as a strong heuristic that there's no chance of finding anything interesting down this road.

• The class–conglomerate distinction is not one of cardinality – it is a distinction of complexity. A class contains sets but may itself be not a set, and in the same way a conglomerate contains classes but may itself not be a class. For example, if $V$ is the universe of sets then even the singleton $\{ V \}$ is a conglomerate but not a class. You don't need a proper class of inaccessibles to model this – one is enough. Sep 20 at 1:12
• @ZhenLin Sure, fair points. The question was aimed at conglomerates isomorphic to no class, and clearly weaker assumptions suffice perfectly well to model such conglomerates. Sep 20 at 5:08