Anyways, since the stuff Agnaa mentioned is cleared up, I guess I can respond to previous points made by other users.
Zeyfil said:
While I fully agree with the dimensional stuff, I can't agree with the 1-A stuff.
Unreachable cardinals are still, in some way, shape, or form, part of infinity, and should therefore still be part of High 1-B (or, at most, a Low 1-A).
I can't imagine any way to keep 1-A and 0 within the confines of math, at all.
Outerverse level isn't really about "transcending the concept of infinity" or something like that, it is just being above the application of spatio-temporal dimensionality, and this is in fact the exact kind of misconception the new system is aiming to fix. Especially with the tier having its primary definition changed to "existing in abstract states so big you cannot reach them by stacking infinities", as it is practice.
Agnaa said:
That is exactly what we do now.
It isn't, though? We give tiers to characters solely based off proposing any higher-dimensional space is automatically infinitely larger, so affecting any infinitesimal part of one is already a higher tier. Given we are going to abandon these practices, it is all but proper we put more emphasis on feats of affecting an area of a given size
Agnaa said:
Those "inconsistencies" only exist because we only assume they're dimensionless/beyond-dimensional in relation to that piece of fiction itself, which doesn't establish that character as being beyond infinite dimensions. This is as much of an inconsistency as "omnipotent" characters appearing in tiers specifically below omnipotence.
We don't even use Omnipotence as an actual concept within our system any longer, and probably never will, so that's a massive false equivalency you got there. The point is that, even if a character is beyond the space-time of a verse with, say, 4 dimensions, they would still exist beyond dimensionality, and thus shouldn't be literally defined as a higher-dimensional being, it's that simple.
DontTalk's points'
DontTalkDT said:
subject (though I already said that noting them as R^n isn't the best idea, as our spacetime is not that by any means)
And as I said, I'm against infinite being demanded, as even for our real life universe it doesn't have to apply. It isn't an arbitrary subset of some larger space of same dimension and if it were it wouldn't matter at all.
It literally is though, a real coordinate space is just a vector space which utilizes the real numbers as the defined coordinates. As I said before, it's literally just Euclidean Space with the notion of a dot product embedded in it.
Yes, hence I already said that we are only using R ^ n in cases where higher-dimensional spaces are either infinite
or defined as infinitely larger. I don't know why you keep hitting the same tile over and over again here when I already clarified that. Even then, we are mostly equating the size of stuff to R^n here, so that just further renders this point moot.
DontTalkDT said:
Don't know where you get that idea from. A tuple can fundamentally be defined as a function from an index set unto the set the values in the tuple come from. Usually the index set is finite, but there is no particular reason it has to be. If you take a sufficiently large index set I and keep the codomain the real numbers, you get a vector space with the number of dimensions we are looking for. The addition and scalar multiplication can be defined basically the same (each index is added separately and is multiplied separately).
It mostly comes from the fact that n-manifolds (a class of objects which encompasses both euclidean spaces and real coordinate spaces,btw) are most commonly defined as fundamentally
second-countable structures, which itself limits the extent of their cardinality to being equivalent to, or smaller than that of the continuum of the reals, as that would mean that the basis which generates their topology has to be strictly countable, and thus at most of cardinality aleph-null, which can obviously only have unions reaching up to aleph-one (assuming CH holds and blah blah)
Anyways, the point is that, if one assumes a manifold of cardinality exceeding R, much of the theorems which normally hold in relation to manifolds as a whole just fall apart altogether, such as the assertion you can embed one within some arbitrary n-dimensional euclidean space. This is the case for most of those things when one assumes the Continuum Hypothesis holds, by the way: in terms of cardinality, they end really early in the aleph hierarchy, as I mentioned up above.
DontTalkDT said:
Keep to ZFC, also terminology wise. Anything else is needlessly confusing
but i dun want to >:c
I don't really see a reason to be so zealous about keeping to ZFC, specially when NBG is literally just an extension of it which allows us to refer to Classes as actual objects within it. All of the theorems which hold in the former also hold in the latter.
DontTalkDT said:
Being outside the arrangements of real and complex numbers has nothing to do with using classes, though. Most sets are.
The reason for the question was that you apparently have a tier for uncountably many higher planes and then define a higher tier via cardinals...
I am well aware of that, the new system mostly uses classes to define this stuff for specificity's sake, as to avoid the idea of aleph numbers and the like having fixed tierings, since they are mostly quantities, and not actual, defined objects, but I believe you already know the gist of that.
DontTalkDT said:
You can't even proof that inaccessible cardinals exist.
Mathematics that are not even well supported by ZFC (it's consistent with ZFC, but you know what I mean) is neither easy, nor intutive, nor can I imagine why in the world you want to do absolutely anything with it in regards to our tiering.
Yeah, but you can still assert their existence as an additional axiom alongside whatever universe of set theory you are using, just like how you don't even have to take into account the axiom of infinity, but still can do such
(especially when you want to have some actual fun in the barren wasteland that is math) to spawn all of the enormous shit that is interesting to talk about. And to be perfectly honest, I don't see how "an uncountably infinite number so big it cannot be accessed from smaller quantities no matter what" is really that hard to grasp, it's literally just to the universe of alephs what ¤ë is to finite numbers.
Equalization, simple as that. Not that many verses make any mention of spatial dimensions either, but we still equate them anyways. Same thing here, more details on why is that being done lie within the comment I posted in response so Sera just above.
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