Von Neumann Universe & Grothendieck Universe scaling

[AnosVoldigoadDemonKing]

Hello everyone, I have a question about the Grothendieck Universe and the Von Neumann Universe.

All I know is that the Von Neumann Universe theory uses sets, but I don't get it at all cause I'm kind of slow at understanding things, and that the Grothendieck Universe theory goes beyond the Von Neumann Universe sets, but what's the difference? And where would the Grothendieck Universe theory scale? To Low 1-A or higher?

 

[Laxxius]

Here's a quick, simplified explanation of how it all works according to my understanding:

Once you get to Tier 1, you start measuring things in terms of some number of dimensions. For example, Low 1-C is 5 dimensions, while 1-C is 6 to 9 dimensions. For the rest of Tier 1 until Low 1-A, everything is measured in terms of some number of dimensions. At and after Low 1-A, the number of dimensions are no longer measured, and other metrics are used instead.

Here's a list of how many dimensions each tier contains:
Low 1-C: 5 dimenions
1-C 6-9 dimensions
High 1-C: 10-11 dimensions

Tier 1-B is a little different, because it doesn't have a normal range of numbers. It is 12 or more dimensions. In other words, any amount of dimensions less than infinity. This shows that a simple range of numbers is not always sufficient to describe the number of dimensions contained in a tier.
Tier High 1-B: infinite dimensions

Since Tier 1-B is already at infinite dimensions, but there are tiers above it that apparently contain a larger amount of dimensions, a new concept needs to be introduced: Different levels of infinity. Basically the idea is that some infinities are bigger than others, which is strange, but it's just some weird, higher level math stuff. In any case, the different sizes of infinity all have different names. The "smallest" infinity is called Aleph-0. The second smallest infinity is called Aleph-1, and I think you can see the pattern.

So, Tier High 1-B is actually Aleph-0 dimensions
And Tier High 1-B+ is Aleph-1 dimensions or more (so think Aleph-2, Aleph-3, Aleph-1,000,000)

Now for your actual question:

As it turns out, it's actually pretty hard to figure out all the different sizes of infinity. We have Aleph-0, Aleph-1, etc., but there are actually infinities that are bigger than all the Alephs, no matter how far into them you go. And then there are infinities bigger than those infinities, and so on.

In order to discover new infinities, Axiomatic Systems are used. There are a bunch of different Axiomatic Systems, and mathematicians can pick their favorite one to use to try and find new, bigger infinities. However, some Axiomatic Systems are "stronger" than others, in the sense that some Axiomatic Systems are able to find bigger infinities than other ones can.

For example, Z_2 is the name of an axiomatic system (yes, they all have weird names), but it's not a very strong one. ZFC is a much stronger axiomatic system, and is the most commonly used one. There is an even stronger axiomatic system called TG. It can use Grothendieck Universes to discover infinities even bigger than the biggest infinity you can discover with ZFC.

Von Neumann Universe, to oversimplify a bit, is basically a list of every possible Axiomatic System you could use. It even has Axiomatic Systems we haven't thought of yet, that could discover even bigger infinities than anything we've found so far. So when Tier Low 1-A says a character is on the level of a Von Neumann Universe, it means the level of infinity they can reach / the number of dimensions they can reach, is essentially Boundless. Or, we don't know of any upper-bound to their power level, because Von Neumann Universe contains Axiomatic Systems that themselves can discover bigger infinities than anything we've found so far.

 

[AnosVoldigoadDemonKing]

Thanks for explaining it, and the Grothendieck Universe would it still scale to Low 1-A?

 

[Laxxius]

Thanks for explaining it, and the Grothendieck Universe would it still scale to Low 1-A?

High 1-B+