Tiering System Revisions - Part 3

[Jockey-1337]

My comment has been deleted, may I ask why?

 

[Nepuko]

It may be best if non-staff stop commenting here, unless they received special permission, as we need to try to make some progress.

@Jockey

 

[Ultima_Reality]

Maffs

Hamel basis allows for countably infinite dimensional spaces, just not Banach spaces.

Schauder basis is not important, as you don't define dimensions over it, as far as I am aware.

I didn't know about that. Assuming that is true, so, then? What's exactly the problem? We are not really using any specific kind of infinite-dimensional space, just assuming an arbitrary structure with that many dimensions, there's no need to be that nitpicky.

You do, though. As far as I know, the Schauder Basis is very much the same as the Hamel Basis, with the difference being that it allows for linear combinations between vectors to be represented as infinite series, whilst the Hamel Basis only allows for finite sums.

We could just define all as power equal to destroying large arbitrary c-dimensional manifold. Though I'm not familiar with the different kinds of infinite dimensional manifolds, so not sure if it's a good idea.

Should be an arbitary space over R at least, otherwise you end up with some weird space over F2 or something.

I guess we could just say manifolds for finite dimensions and large infinite dimensional structures beyond that.

I am... really not fine with saying any infinite-dimensional space has dimension equal to the continuum. I am aware the system I propose goes neck-deep into mathematics, but they are streamlined enough to only apply to the higher parts of the system. There is no need to nitpick lower parts to that extent.

Just define an arbitrary infinite-dimensional space and call it a day, as I've already said.

Is second countable even required for infinite dimensional manifolds? Not an expert on that topic, but by the general definition I find on the infinite dimensional case it isn't.

I would find it strange if Hilbert space structure is not as good as Manifold structure, as it is generally much more well behaved...

I don't see why it wouldn't. I do know that there are infinite-dimensional spaces that have the property of second-countability on them, such as the aforementioned R^N.

See above, we are taking liberties here and not choosing any specific infinite-dimensional space. This system is about size.

The important part is that it's Hausdorff for that. As said, many not second-countable topologies are metrizable. You can get a metric on any given number of dimensions.

Yeah, that's what I said. If a given space is not Hausdorff nor satisfies any of the separation axioms, then it is not metrizable in the first place. I mentioned second-countability due to that property's additional relation to manifolds and metrizability in the first place, although I guess you already saw my answer in the edit I made.

Although, I would like to correct myself on one thing: Namely, the fact that spacetime is actually modelled as a pseudo-riemannian manifold, which makes it a pseudometric space instead, so it's not riemann. But, that's really minor, since the space as a whole still needs to be hausdorff and satisfy the separation axioms, so one can actually define any point in time as being separate from one another, plus the other reasons I mentioned still hold. So, semantics too, sorta.

Anyways, I reeeeally don't wanna drag this argument for too long and the points have already been made for the sake of saying this stuff is indeed mathematically plausible, as far as I see. At this point we could just say we are defining some abstract set of objects outside of spacetime which has cardinality P(P(N)) or something, and it would be good, in my opinion.

What does "reproducing physical spacetime" mean, though? You can have things like distance and angles.

You could abstractly define distance past a certain point, but I don't think you could so such in a weird space which doesn't satisfy any separation axiom and is neither metrizable nor uniformizable. Again, see above.

In regards to the edit you made to the last point I can just say that it was only meant as counter example, not as a suggestion for what to use. The point is that second countable topologies exist regardless of cardinality, meaning that we can't exclude there being suitable topologies due to that criteria at least.

Yeah, but only under spaces defined under the Indiscrete Topology, a property which Manifolds (the stuff which dimensions are defined under, for the most part) cannot have by definition. Sure, Second-Countability is a constant that can exist regardless of cardinality in the context of trivial topologies, but when we are dealing with Manifolds and stuff with different properties which have nothing to do with indiscrete spaces, then it has a very strict upper bound when it comes to its cardinality.

So all of this comes down to semantics, really.

If we are at the point of how to define things similar to physical spacetime: Can we even consider any countably dimensional spaces as such? Most strong assumptions about the nature of such a space would give us something like an incomplete space. So, if we have a character destroy an entire infinite dimensional space, is it plausible to assume it is incomplete?

Again, I really don't see why we need to be this nitpicky in relation to fiction: To paraphrase myself: I am aware that the system I am proposing uses Mathematics in the first place, but there must be nuance here, and it is streamlined so the bulk of it only applies to the higher tiers. Starting to get picky over this stuff will lead nowhere, especially when we get to heavy assumptions which don't at all fit with most of fiction such as "every infinite-dimensional banach space is necessarily uncountably infinite-d". You should remember we are basing this off of size.

But, for the sake of the argument: I don't see why this really matters, if an infinite-dimensional space is a complete metric space, then it is a banach space by consequence, and I've already stated that there can indeed be countably-infinite dimensions in such a space if we move away from the Hamel Basis and tackle other notions.

Nevertheless, this argument has devolved into being mostly semantics and needless nitpicking, so I say it is best we move on and make a new thread by now, pls. We need to make progress.

 

[Andytrenom]

4 parts yay

How much of it was even discussing the actual logic and not presentation. Not that I helped that in any way

 

[Antvasima]

@Ultima

Feel free to do so.

 

[Magi_Hussie]

Welp, I guess it's safe for me to pick option 3 then since its the one I can mostly agree with.

 

[Agnaa]

Starting to get picky over this stuff will lead nowhere, especially when we get to heavy assumptions which don't at all fit with most of fiction such as "every infinite-dimensional banach space is necessarily uncountably infinite-d". You should remember we are basing this off of size.

It seems weird to say that assumption doesn't fit with most of fiction, when there's maybe 3 or 4 pieces of fiction on the site at most (and 0 that I know of) that actually distinguish between countably infinite-d and uncountably infinite-d.

 

[Ultima_Reality]

I wasn't really referring to the existence of uncountably infinite-dimensional spaces in general, but more like the notion that -all- infinite-dimensional spaces must have uncountably infinite dimensions.

 

[Agnaa]

Yes, but what I'm saying is simply: Almost no fiction specifies that countably infinite-d spaces aren't necessarily uncountably infinite-d.

So your claim that "most of fiction" doesn't fit with that doesn't hold up. In fact, almost no fiction makes that distinction.

 

[Ultima_Reality]

Yes, but the difference between countable and uncountable infinity is stupidly big, and most authors would likely mean a basic, numerical infinity when mentioning a space with infinite dimensions, more often than not. Assuming uncountably infinite dimensions is a ridiculously big high-end for anything, really, so we should assume the lowball lest we end up wanking a bunch of characters to the high heavens based on nothing.

 

[Antvasima]

Ultima is probably correct.

By the way, we are almost out of posts, so it would be a good idea to start a continuation staff discussion thread very soon.

 

[DontTalkDT]

That feeling when you write a long post, but forget to copy it before posting and wikia swallows it. RIP. In that regards I can't wait for the new forum...

Ok, then I will make this much much shorter. Might be better that way. I will get back to correcting the complicated stuff if it becomes relevant.

Just define an arbitrary infinite-dimensional space and call it a day, as I've already said.

I mean, we can just say "destroying a c-dimensional space" for each tier.

But if you want to use arbitrary spaces then you will need to reformulate the part of your justifications that assumes manifolds, second countable topological spaces and other notions that assume very specific kinds of space.

(Personally I also don't understand the standpoint of formulating a system using deep mathematics and then not wanting mathematical consequences, but ok)

 

[MasterOfArda]

That feeling when you write a long post, but forget to copy it before posting and wikia swallows it. RIP.

The world is a dark place.

 

[Iamunanimousinthat]

That feeling when you write a long post, but forget to copy it before posting and wikia swallows it. RIP.
The world is a dark place.

I find pressing undo on ya keyboard gets it back.

 

[Ultima_Reality]

That feeling when you write a long post, but forget to copy it before posting and wikia swallows it. RIP. In that regards I can't wait for the new forum...

I am extremely sorry for your loss.

I mean, we can just say "destroying a c-dimensional space" for each tier.

Well, ****, I thought you meant c as in, the continuum, my bad. My bad2 if that's actually what you meant

(Personally I also don't understand the standpoint of formulating a system using deep mathematics and then not wanting mathematical consequences, but ok)

I am not trying to avoid mathematical consequences, though, it's just that those consequences have little to do with what the proposed system is trying to accomplish. It's mostly attempting to tier the size of layers / dimensions through a stable metric (in a way that reasonably fits with fiction) and approach a way to solidly map out the higher tier stuff as I've explained, unless your diseased post has a few points I'd like to address.

What I meant, is that we don't necessarily need specific examples of spaces past a certain point; at High 1-B, we could as well define an arbitrary countably infinite-dimensional manifold and use it to approximate the size of similar spaces. That's the most we really need to do, imo.

 

[Antvasima]

Just a reminder that we only have 33 posts left here now. A new thread would be very appreciated at this point.

 

[Ultima_Reality]

I can make it tomorrow, assuming that someone else doesn't do it first, that is.

 

[Antvasima]

Okay. Thanks.

 

[Ultima_Reality]

Okay, just so we are all on the same page. What exactly should I summarize in the next thread? It should be the final part of these discussions, so I don't think going towards the "muh options" path is really the best idea at this point, as in all likelihood it'll end up like the previous ones. Should we just do what I suggested a few posts ago? As in, analyze the 1-A verses first and then choose what is more optimal for our purposes?

 

[Antvasima]

I think that selecting what alternative that we should go with is necessary to include, preferably based on the list that I kept quoting earlier.

We do already seem to have decided to use uncountable degrees of infinity or equivalent for Low 1-A though.

I am not sure whether or not it is realistic to diverge into evaluating all of the 1-A verses within the thread.

 

[Ultima_Reality]

Not really saying we should suddenly start analyzing every single 1-A verse in a single thread. Just noting that it may serve as a place for we to discuss how we are gonna do that and figure out the kinks of it. And to be perfectly honest, it seems like the best solution to me, since the logic behind the proposed system has already been debated to death, most of the original options are essentially dead and buried, and from what I've gathered, the people who actually gave their inputs and care to a small degree are mostly fine with it, with a few exceptions here and there which have been already addressed (As far as I remember anyways).

If you deem this to be a bad idea, I guess we can just default to this for convenience's sake, and then decide to start analyzing verses based on it. That way people can evaluate them properly using a solid ground and tweak it a bit if anything of note happens:

High 1-B = Countably infinite dimensions / layers = N

Low 1-A = Uncountably infinite dimensions / layers at their lowest bound = P(N)

1-A = Abstract settings with size roughly analogous to further uncountable sets of layers = P(P(N)), P(P(P(N))), P(P(P(P(N)))), and so on = Outerverse level and finite number of steps above it.

1-A+ = Infinite steps above baseline Outerversal

High 1-A = Above any recursions of size on any previous scale.

0 = blahblah transcendence / perspective

I also love how we thoroughly neglected to discuss what verses will be affected by the changes regarding how dimensions are going to be applied and decided to discuss the presentation of 1-A stuff instead.

 

[Antvasima]

1) Okay. I am fine with your suggestion then.

2) That could be an idea, but Matthew suggested that your 1-A+ should be High 1-A, and your High 1-A should be tier 0.

3) We should probably discuss how dimensions will be applied more, yes.

 

[Matthew_Schroeder]

What I said was that High 1-A should just be anything above infinite Outerversal levels. And the blah blah Transcendence / boundless stuff should be Tier 0

Also the next thread should hopefully be focused on the stuff we've been ignoring rather than 1-A semantics.

 

[Antvasima]

Okay.

 

[Nepuko]

I am fine with Ultima's suggestion above .

 

[TheUnshakableOne]

I had a couple of questions about the dimensional side of things that i saw on the first thread.... is this a good moment to ask them...? or should i wait a bit longer..? I am also sorry to interrupt you guys....

 

[Antvasima]

We have barely any posts left, so now is not a good time, no.

 

[The_2nd_Existential_Seed]

I know I shouldn't be responding but this will be my only reply to this thread: I think it'd be best to settle the 1-A issue here and now with siding with Ultima's statement about the 1-A stuff then focus on the other stuff (such as verses affected) in the next thread exclusively.

 

[Antvasima]

We only have 17 posts left. Immediately stop responding, and wait for Ultima to create a restart thread.

 

[Ultima_Reality]

https://vsbattles.com/vsbattles/3612613

 

[Antvasima]

Thank you. I will close this thread and highlight the new one.