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

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    $\begingroup$ By "conglomerates" do you mean proper classes? $\endgroup$ Sep 19 at 17:27
  • $\begingroup$ @MichaelHardy No; I mean the next size up. $\endgroup$
    – anon
    Sep 19 at 17:29
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    $\begingroup$ You mean next after proper classes? $\endgroup$ Sep 19 at 17:29
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    $\begingroup$ @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. $\endgroup$ Sep 19 at 17:41
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    $\begingroup$ Well, why was the definition of conglomerate introduced in the first place, and by whom? $\endgroup$
    – fosco
    Sep 19 at 18:05
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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.

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  • $\begingroup$ The surreal numbers are a really nice field and its a proper classes. $\endgroup$
    – anon
    Sep 19 at 21:45
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    $\begingroup$ @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. $\endgroup$ Sep 19 at 21:47
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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.

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  • $\begingroup$ 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. $\endgroup$
    – Zhen Lin
    Sep 20 at 1:12
  • $\begingroup$ @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. $\endgroup$ Sep 20 at 5:08

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