Beyond the Von Neumann Universe

[ArachDusa]

This post is going to kind of be two questions about the same general subject, which is mathematical structures that surpass baseline Low 1-A, or in other words, surpass the Von Neumann Universe. The wiki's tiering system page briefly mentions "the Von Neumann Universe (And larger structures still)", with the only context being a hyperlink to a Wikipedia article about mathematical "conglomerates", which are... A bunch of convoluted math jargon that's way over my head. However, the one thing I did understand is that conglomerates are a feature of various set theories, including ZFC and TG set theory. I heard someone once say that ZFC is greater than the Von Neumann Universe and TG is greater than ZFC, but when I looked it up, I found this thread where wiki user Laxxius debunks this claim by stating that the Von Neumann Universe encompasses every possible set theory/axiomatic system. Normally I would chalk it up to a disagreement or factual error, but as mentioned, the wiki's own tiering system page places a single element of an aspect of a specific set theory as a larger structure than the entire Von Neumann Universe. So what's going on there? Am I missing something, or was Laxxius just wrong? Can anyone explain this without sounding like gibberish to people who don't understand all this complex math?
...That part was longer than I expected. But I do have another question that's kind of related. In the past, Hoyoverse Aeon profiles listed the imaginary tree as Low 1-A with a statistics value of "Absolute Infinity - Ω", again, without added context. This rating was letter removed and replaced with a note that "absolute infinity" in this context does not refer to Cantor's absolute infinity, but this removal appears to have recently been undone without updating the statistics value as of this writing. From this I can infer that an absolute infinity should also be Low 1-A, but the fact that it is (Or was originally) a statistics value implies that it's different from the Von Neumann Universe or the baseline of the tier. My intuition tells me that it would be the highest direct value possible for Low 1-A and the only way to be higher into the tier would just be through upscaling, but I wouldn't be surprised if I'm completely off the mark. So my second question is, how far into Low 1-A would absolute infinity be?

 

[Vietthai96]

1. Von Neumann Universe is the largest, encompassing every possible set theory/axiomatic system, as already explained in the thread you linked

2. Absolute Infinity was once considered as Low 1-A in case of mathematical absolute infinity, but now it isn't anymore as explained by Ultima here, so it is either untierable or Tier 0 based on Cantor's True Absolute Infinity which is his view on God

 

[ArachDusa]

1. Von Neumann Universe is the largest, encompassing every possible set theory/axiomatic system, as already explained in the thread you linked

Then why does the wiki's tiering system page explicitly mentions conglomerates as larger than the Von Neumann Universe?

 

[SweetDao]

Then why does the wiki's tiering system page explicitly mentions conglomerates as larger than the Von Neumann Universe?

Layer into Low 1-A really

 

[Ihsjihahxu]

Then why does the wiki's tiering system page explicitly mentions conglomerates as larger than the Von Neumann Universe?

CLASSES

To handle "large collections" such as "all sets", we require that:

(1) For each "property" P we can form a "collection" whose members are exactly those sets that have property P. We call this the class of all sets with property P and denote it by (xx is a set and P(x)) (or more briefly, by {x | P(x))). For example, we can form the "class of all sets", the "class of all ordinal numbers", and the "class of all groups". Obviously, classes are precisely the "subcollections" of the class of all sets. We will call V the "universe".

(2) For convenience of expression, we also wish to regard sets as special classes.

(This can easily be achieved by adding to the above requirements for sets, the additional requirement that each member of a set be a set.)† Those classes which are not sets are called proper classes. Often sets are referred to as small classes and proper classes are called large classes. This distinction between "small" and "large" will turn out to be essential in many categorical investigations. Notice that the universe % of all sets is a proper class and that Russell's paradox now translates into the harmless statement that the class of all scts that are not members of themselves, is not a set but is a proper class

CONGLOMERATES

If A is a proper class, then there exists no class that has A as a member (since every member of a class must be a set). However, we will occasionally need to consider "collections" of classes. For this reason, we introduce the broader concept of "conglomerate". Roughly speaking, conglomerates are

+ This also implies that if the pair (x, y) is a set, then so are x and y."collections" having classes or conglomerates as members. In particular, we require that:

(1) Every class is a conglomerate.

(2) Conglomerates are closed under the usual set-theoretic constructions outlined above (1.1); i.e., they are closed under the formation of pairs, unions, products, etc.

Thus we can effectively treat conglomerates in the same manner that we treated sets; we can construct functions between them, equivalence relations on them, etc. There is the temptation, of course, to form the "cartel" of all conglomerates. However, assuming that our primary interest lies with "usual categories", such as the category of all sets or the category of all topological spaces, we will not need to consider such an entity.

N or R → are sets
all sets (called the universe V)
The class of all groups→ are large classes ( Proper Clas )

If we need to collect classes into a collection → conglomerates

 

[DoFliesCallUsWalks]

Actual Infinity, which has similar immutability traits as tier 0, and then Logical pluralism, then all the self-invented shit that's beyond the large cardinals and zfc axioms. You can pile some philosophy or anti-logic stuff on top but that just gets you further and further into H1A