Low 1-A

[Voidnether]

Yeah, I previously concurred with tiering Absolute Infinity as Low 1-A out of an equation to a proper class but I realize the above is just correct. A proper class includes all sets, but within the framework of set theory itself you can go, in a sense, "further" than them. You can make products of proper classes (Like ORD², which is the class of all 2-tuples of ordinals) and in category theory you can extend that to conglomerates thereof, and so on and so forth. Meanwhile "Absolute Infinity" in the naive set theory sense is applying the definition "Set of all sets" to an unqualified notion of "set" as meaning any collection whatsoever. Meaning it includes everything, including itself (Hence it's a paradoxical object). Nothing is further than it.

So wrt to Absolute Infinity you either: 1) Try to make it quantitative and so arrive at an impossible (re: untierable) object, because the "quantity of all quantities" isn't a thing. 2) Try to go with Cantor's theological view of it and argue it's Tier 0 (Which obviously can't happen here).

I remember there are 4 things that qualify as Low 1-A back then which is:
1. Von Neumann Universe
2. Multiverse Type 4
3. Absolute Infinity
4. Set of all sets

Given Absolute Infinity does not qualify for Low 1-A due to it being treated as an universal set here, would it also apply to set of all sets too?

 

[SweetDao]

Given Absolute Infinity does not qualify for Low 1-A due to it being treated as an universal set here, would it also apply to set of all sets too?

Set of all sets can't exist, so you're effectively talking about the class of all set and that would be Von Neumann universe, basically.

 

[Voidnether]

Set of all sets can't exist, so you're effectively talking about the class of all set and that would be Von Neumann universe, basically.

I see, I was confused for a little cause Ultima said this when set of all sets in the first place is just an universal set so I don't get the part about him saying that Absolute Infinity is applying the definition of "set of all sets" to an unqualified notion of set when set of all sets is literally untierable so I thought it would still qualify (for the set of all sets one)

 

[SweetDao]

I see, I was confused for a little cause Ultima said this when set of all sets in the first place is just an universal set so I don't get the part about him saying that Absolute Infinity is applying the definition of "set of all sets" to an unqualified notion of set when set of all sets is literally untierable so I thought it would still qualify (for the set of all sets one)

The set of all sets can't exist coherently because how can you have "the box of all boxes" not be inside the denomination of "all boxes" in the first place? That creates some sort of self-referential paradox where the "the box of all boxes" include itself while being outside of it, somethat's that's not possible, hence why only a class would work to "contain all the boxes", in that example.

That's why he said you could only have two outcomes. Either the quantitative one (the above example shows it just can't work without twisting logic) or the Theological one.