On the Many, Many Incoherences of the Tiering System

[DontTalkDT]

How exactly to interpret topological properties in terms of physical phenomena is not an easy question.
What can be objectively said is that in a non-Hausdorff space limits are not unique, which means that things like unique derivatives likewise don't exist. That means that derivative-related concepts, such as speed, can't be defined as usual. They generally become "fuzzy".

Not sure if that helps your debate along. What is the overarching reason for the question? If you're considering such abstract properties there are likely a number that anyone would dearly miss. A non-Hausdorff space for instance already is not metric, which might put them beyond what the average person consider dimensional space. Math also has like a dozen different notions of "dimensionality".

 

[Agnaa]

I find it odd that you mention "People noticing" things, though. Obviously, people would notice the difference between a 3-D space and a 4-D space, and yet that wouldn't constitute a breach in the common conception of dimensionality to you. At that point I'd ask what exactly it is you think the average person even conceives of when they think of spatial dimensions as a concept.

Axes of space which function largely similarly to the ones IRL.

It's pretty much identical, I might note. Since "4-dimensional + Non-Euclidean" follows exactly the same format as "Has more than c dimensions + Non-Separable." That format being "Bigger than what came before it + Has a property that makes it kinda weird compared to the things it dwarves."

I don't think "No matter how many dimensions you add, this character would be above them" is accurately represented by being 4-dimensional, for reasons which I hope are obvious.

That doesn't really change my point. This being that you seem to view even things like "This void is timeless, spaceless and beyond dimensions, and is depicted as surrounding and infinitely dwarfing dimensional space" in terms of some Dragon Ball-styled thing where the void simply has "More power" than it things it dwarves in some nebulous ill-defined sense, and since "power" to you is measured in volume, then that can be equated to volume even if the void itself has no volume to speak of.

It isn't, why do I have to keep saying this?

I'm saying that's not really coherent. If the hypothetical beyond-dimensional void's non-quantitative "size" is one and the same with its "power," then its power likewise isn't a quantitative thing (By transitivity), and can't be equated to a quantitative measure such as volume. The other arguments I've done follow suit after this.

I don't agree with drawing such a difference. I thought we'd kind of come to an agreement on that before, at least in the extreme, with vastly different axiomatic systems still being "quantitative".

Not exactly sure what you mean by that. The basis of the Tiering System below 1-A would be just dimensional jumps, and the basis of the Tiering System from 1-A and onwards would be the aforementioned notion of "meta-size." Both of those would simply be grounded on the basic notion of "X is bigger than Y," which in fiction has to extend further than simple comparisons of volume and measure. It's simple, really.

Then we're talking about the notion of "X is bigger than Y in a manner which extends further than simple comparisons of volume and measure".

Why do I have to keep saying "I'm using whichever general notion for tiering you're using", while you dance around the idea of such a thing actually existing?

 

[Agnaa]

How exactly to interpret topological properties in terms of physical phenomena is not an easy question.
What can be objectively said is that in a non-Hausdorff space limits are not unique, which means that things like unique derivatives likewise don't exist. That means that derivative-related concepts, such as speed, can't be defined as usual. They generally become "fuzzy".

Not sure if that helps your debate along. What is the overarching reason for the question? If you're considering such abstract properties there are likely a number that anyone would dearly miss. A non-Hausdorff space for instance already is not metric, which might put them beyond what the average person consider dimensional space. Math also has like a dozen different notions of "dimensionality".

The overarching reason for this question is:

Agnaa: The definition of "beyond dimensions", when sufficiently evidenced, should be based on spaces beyond those that are separable and Hausdorff. This is since dimensions work that way IRL, and those qualities confer certain properties that people would notice and find the absence of strange. Thus, they're implicitly part of the definition when most people talk about things like that.

Ultima: The definition of "beyond dimensions", when sufficiently evidenced, should be based on spaces describable by Cartesian products, which includes arbitrarily-large cardinals in most extensions of ZFC. This is since people IRL only think of "dimensions" in terms of "directions", and that's a simple, common definition that gets at the essence of that. On the contrary, spaces being separable and Hausdorff are mathematical curiousities that few would notice.

I don't know enough about what those properties imply to be 100% confident that their absence would be noticeable to people. I'd expect they would, but Ultima said they definitely wouldn't, so I seek clarification from another person knowledgeable on those sorts of things; you.

 

[DontTalkDT]

The overarching reason for this question is:

Agnaa: The definition of "beyond dimensions", when sufficiently evidenced, should be based on spaces beyond those that are separable and Hausdorff. This is since dimensions work that way IRL, and those qualities confer certain properties that people would notice and find the absence of strange. Thus, they're implicitly part of the definition when most people talk about things like that.

Ultima: The definition of "beyond dimensions", when sufficiently evidenced, should be based on spaces describable by Cartesian products, which includes arbitrarily-large cardinals in most extensions of ZFC. This is since people IRL only think of "dimensions" in terms of "directions", and that's a simple, common definition that gets at the essence of that. On the contrary, spaces being separable and Hausdorff are mathematical curiousities that few would notice.

I don't know enough about what those properties imply to be 100% confident that their absence would be noticeable to people. I'd expect they would, but Ultima said they definitely wouldn't, so I seek clarification from another person knowledgeable on those sorts of things; you.

Well, if they are non-Hausdorff then they are non-metric which is definitely very noticeable. Although I'm not sure how you arrived at those two properties in particular.

 

[Agnaa]

Well, if they are non-Hausdorff then they are non-metric which is definitely very noticeable. Although I'm not sure how you arrived at those two properties in particular.

They were the ones mentioned when the current broad strokes of the tiering system were introduced in 2019, as justifying the placement of 1-A. At the time, they were said to be the point where conventional space-time dimensions end, as agreed by you, Ultima, and Aeyu, iirc.

 

[DontTalkDT]

They were the ones mentioned when the current broad strokes of the tiering system were introduced in 2019, as justifying the placement of 1-A. At the time, they were said to be the point where conventional space-time dimensions end, as agreed by you, Ultima, and Aeyu, iirc.

Hmmm... I think I recall that Ultima brought up some abstract topological properties back then for some reason, yeah. Think the details might have been somewhat different, but what exactly happened probably isn't too important to your debate.

 

[Agnaa]

Well then, I guess to try to re-rail things a bit, Ultima, do you agree or disagree with DT's statement here?

Well, if they are non-Hausdorff then they are non-metric which is definitely very noticeable.

 

[Ultima_Reality]

They were the ones mentioned when the current broad strokes of the tiering system were introduced in 2019, as justifying the placement of 1-A. At the time, they were said to be the point where conventional space-time dimensions end, as agreed by you, Ultima, and Aeyu, iirc

Yeah, that was not a particularly good decision (And DT quite literally forgot that this was even a thing last time it was brought up in the Low 1-A thread, months ago). The logic was essentially that the topology of the spacetime manifold as usually formulated in physics happens to have the properties of both Separability and Hausdorffness, and since a set with these two properties has its cardinality cap at 2^c, then anything past 2^c is already "beyond conventional spacetime dimensions."

That's a fairly shoddy argument for a number of reasons. The first and main one being that "space dimensions" at best can be defined as "A continuum of numbers with which physical locations are specified," and temporal dimensions cling onto that due to being just a generalization of spatial ones. So, obviously, neither of them meet their cap at 2^c and nor do they particularly care about either of the two properties.

Another reason is that Separability and Hausdorffness aren't actually particularly important when defining dimensionality (No definition of "dimension" in math cares about those two). Separability literally just means "A space X has some subset Y that's countable and has its members be all either those of Y or arbitrarily close to those of Y," which doesn't have immediate ties to the physical stuff (Which is what we'd be interested in).

Meanwhile, a space X being "Hausdorff" means "Every given point in X has a set around it that's disjoint from the sets that each contain the other points." Basically just "Every point in X is separated from other points by some radius," so without it you can have two points, x and y, that are distinct but nevertheless have 0 distance between them.
As I said before, that's obviously closer to the mind of a layman than the notion of separability, but ontop of the first reason, it doesn't justify the above logic because it's not really a property that just... ceases to exist, past a certain size. Any arbitrary product of a hausdoff space is still hausdorff, so R^κ where κ is aleph_3 is still a Hausdorff Space, for instance. And so is any further generalization of that.

 

[Agnaa]

Damn, so you'd need both for that cap, but you both seem to agree that one of them isn't really relevant. Fair enough.

My only remaining concern with that, then, is that you'd apply an amount of scrutiny to whether verses are "truly beyond dimensions" that I'd find insufficient (based on our previous conversations about what qualifies for such things). But ultimately, I guess that'd be my problem for not participating in threads evaluating tier 1 verses enough to act as a counterbalance.

Is there any particular topic you'd like to go to next?

 

[DontTalkDT]

Yeah, that was not a particularly good decision (And DT quite literally forgot that this was even a thing last time it was brought up in the Low 1-A thread, months ago). The logic was essentially that the topology of the spacetime manifold as usually formulated in physics happens to have the properties of both Separability and Hausdorffness, and since a set with these two properties has its cardinality cap at 2^c, then anything past 2^c is already "beyond conventional spacetime dimensions."

That's a fairly shoddy argument for a number of reasons. The first and main one being that "space dimensions" at best can be defined as "A continuum of numbers with which physical locations are specified," and temporal dimensions cling onto that due to being just a generalization of spatial ones. So, obviously, neither of them meet their cap at 2^c and nor do they particularly care about either of the two properties.

Another reason is that Separability and Hausdorffness aren't actually particularly important when defining dimensionality (No definition of "dimension" in math cares about those two). Separability literally just means "A space X has some subset Y that's countable and has its members be all either those of Y or arbitrarily close to those of Y," which doesn't have immediate ties to the physical stuff (Which is what we'd be interested in).

Meanwhile, a space X being "Hausdorff" means "Every given point in X has a set around it that's disjoint from the sets that each contain the other points." Basically just "Every point in X is separated from other points by some radius," so without it you can have two points, x and y, that are distinct but nevertheless have 0 distance between them.
As I said before, that's obviously closer to the mind of a layman than the notion of separability, but ontop of the first reason, it doesn't justify the above logic because it's not really a property that just... ceases to exist, past a certain size. Any arbitrary product of a hausdoff space is still hausdorff, so R^κ where κ is aleph_3 is still a Hausdorff Space, for instance. And so is any further generalization of that.

I wish to note that, for infinite dimensional spaces, you also can't simultaneously have completeness and a notion of distance that extends our finite dimensional notion of distance. I.e. as far as you can construct something resembling physics in infinite dimensional spaces at all, it would be highly alien to a human, even if they only view a self-contained 3D slice. E.g. from a human perspective speed my be direction-dependent.
Likewise, the concept of size (measures) doesn't really cleanly generalize to infinite dimensional spaces. While you can have measures, they are stuff like probability measures which don't really quantify size in a physical sense. Size includes concepts like length, of course.

 

[Ultima_Reality]

Damn, so you'd need both for that cap, but you both seem to agree that one of them isn't really relevant. Fair enough.

My only remaining concern with that, then, is that you'd apply an amount of scrutiny to whether verses are "truly beyond dimensions" that I'd find insufficient (based on our previous conversations about what qualifies for such things). But ultimately, I guess that'd be my problem for not participating in threads evaluating tier 1 verses enough to act as a counterbalance.

Is there any particular topic you'd like to go to next?

I'll make one last post concisely summing up all the points I've made to you so far (And replying to what came above, as well), since I think there are still misunderstandings going about. Whether we come to an agreement on stuff or not, I believe that then will be the time to summarize our general points and get to voting.

 

[Agnaa]

I'll make one last post concisely summing up all the points I've made to you so far (And replying to what came above, as well), since I think there are still misunderstandings going about. Whether we come to an agreement on stuff or not, I believe that then will be the time to summarize our general points and get to voting.

I strongly disagree. There's still a fair few things that I don't think are fully explored.

We kinda unceremoniously dropped the talk about whether R>F differences should be equalised to being beyond all dimensions.

And we haven't talked at all about tiers other than 1-A. Particularly, what to do with the stuff that would now be below it, how to deal with High 1-A/0, and how to deal with tier 11.

 

[Ultima_Reality]

We kinda unceremoniously dropped the talk about whether R>F differences should be equalised to being beyond all dimensions.

And we haven't talked at all about tiers other than 1-A. Particularly, what to do with the stuff that would now be below it, how to deal with High 1-A/0, and how to deal with tier 11.

R>F and Type 2 BDE are interlinked in here, argumentation-wise. The points I'm making for the latter also apply to the former (Which you'll see in my following post, and have seen in previous posts of mine), so I don't think that matters too much here, unless you've other unspoken concerns.

As for the rest: That's true, but I think that at this point it's good to break the topic into chunks. This thread can be left for deciding on the basic concept of "Qualitative superiorities are above any and all quantitative superiorities. Here's what qualifies for qualitative superiority, also" + Giving people a look of what will come in future threads should this one be accepted. Shoving all the specifics and consequents of the whole revision into this single thread isn't a practical solution at all.

 

[Agnaa]

As for the rest: That's true, but I think that at this point it's good to break the topic into chunks. This thread can be left for deciding on the basic concept of "Qualitative superiorities are above any and all quantitative superiorities. Here's what qualifies for qualitative superiority, also" + Giving people a look of what will come in future threads should this one be accepted. Shoving all the specifics and consequents of the whole revision into this single thread isn't a practical solution at all.

I'm fine with splitting to some extent, but I think forking off a foundational concept like "Should there be a tier for things beyond size, such that every character at this tier has the exact same AP" is a bit weird.

I think a split of "foundational concepts to base the system off of", "exact organisation of those into tiers", and "re-evaluation of verses" makes the most sense.

But I guess it doesn't matter too much.

 

[Ultima_Reality]

I'm fine with splitting to some extent, but I think forking off a foundational concept like "Should there be a tier for things beyond size, such that every character at this tier has the exact same AP" is a bit weird.

As I see it: After this thread's subject matter is dealt with, the subsequent thread I make will be about the concept of Tier 0 in light of the proposed 1-A standards. So it would essentially be just Part 1 and Part 2 of the same discussion, whereas the other threads are things to be discussed after the discussion has been wrapped up. Overall, the air in the room's long stagnated, so I think a follow-up thread is the most pragmatic solution for us to have some tangible progress here.

 

[Agnaa]

Well then, I guess I'll pull out the notes with outstanding queries I had for these 1-A changes then.

Would you consider a verse that concretely establishes mathematical hierarchies as encompassing and dwarfing R>F or other qualitative hierarchies to be nerfed to below 1-A, or would you allow math to supersede R>F in that case, going against the definition of the tiers in the first place?

Do you think there would be any hierarchies which aren't definitively quantitative (as in, explicitly mathematical/dimensional), but which also aren't definitively qualitative (as in, based upon real-ness or being beyond dimensionality/mathematics)? I'm expecting things like "higher levels of reality" or "new power systems where each step in it completely dwarfs the previous power system, no matter how far it's extended" might land there, but I'd want to be sure for another set of queries.

 

[Ultima_Reality]

Honestly I'll hold off on writing the aforementioned post for the moment. These questions are way more interesting.

Would you consider a verse that concretely establishes mathematical hierarchies as encompassing and dwarfing R>F or other qualitative hierarchies to be nerfed to below 1-A, or would you allow math to supersede R>F in that case, going against the definition of the tiers in the first place?

That's like asking "What would you do if a verse depicted a thing that explicitly has 5 spatial dimensions as dwarfing something previously described as existing above dimensionality itself?". The answer is obviously that, if that is the case, the latter thing was just never beyond dimensionality to begin with. I'd obviously place those verses below 1-A.

Do you think there would be any hierarchies which aren't definitively quantitative (as in, explicitly mathematical/dimensional), but which also aren't definitively qualitative (as in, based upon real-ness or being beyond dimensionality/mathematics)? I'm expecting things like "higher levels of reality" or "new power systems where each step in it completely dwarfs the previous power system, no matter how far it's extended" might land there, but I'd want to be sure for another set of queries.

No such thing as a middle ground between the two, no. Either you have evidence of constituting a qualitative superiority, or you don't, and are assumed to be quantitative by default. And that's assuming superiority is even indicated at all, of course.

Those hypotheticals you bring up have issues with them, also, because they're completely unqualified statements that can be taken in a myriad ways.
For example I could take "higher level of reality" to mean levels of realness, or I could interpret "reality" as an object that has levels (In which case those levels needn't have any superiority/inferiority relation between each other at all). "new power systems where each step in it completely dwarfs the previous power system, no matter how far it's extended" is also entirely dependent on how exactly this "extension" of the previous power systems happens. I can conclude nothing from those, in short.

 

[Ultima_Reality]

And on a less important note:

I wish to note that, for infinite dimensional spaces, you also can't simultaneously have completeness and a notion of distance that extends our finite dimensional notion of distance. I.e. as far as you can construct something resembling physics in infinite dimensional spaces at all, it would be highly alien to a human, even if they only view a self-contained 3D slice. E.g. from a human perspective speed my be direction-dependent.
Likewise, the concept of size (measures) doesn't really cleanly generalize to infinite dimensional spaces. While you can have measures, they are stuff like probability measures which don't really quantify size in a physical sense. Size includes concepts like length, of course.

The second claim is only a half-truth. I assume you're referring to the common claim that there is no infinite-dimensional analogue of the Lebesgue Measure. For an explanation of this, I draw attention to what you said: It doesn't cleanly generalize, but it does generalize.

Basically, for those who don't know: The "Lebesgue Measure" is the standard notion of size that we use to measure solids in math. So, when we talk about the length of an object, or its area, or volume, and so on, we're grouping all that under the Lebesgue Measure. So we say "Area" is the 2-D analogue of it, "Volume" is the 3-D analogue of it, and so on.

When we jump to infinite-dimensional space, though, you often see the claim that there isn't any analogue of it there, which, as said, is only a half-truth. Math people say that because they care very much about a space having a subset of finite size (measure), since most analysis happens on the level of the finite, for obvious reasons. And so, the reason it's often said that "There is no infinite-dimensional analogue of the Lebesgue Measure" is:

Let (X, ||·||) be an infinite-dimensional separable Banach space. Then the only locally finite and translation-invariant Borel measure μ on X is the trivial measure, with μ(A) = 0 for every measurable set A. Equivalently, every translation-invariant measure that is not identically zero assigns an infinite measurement to all open subsets of A.

In English, this translates to "The only measure that assigns finite sizes to any part of an infinite-dimensional space is the measure that says every subset of that space is of size 0. Otherwise the only available measures are ones that assign a size of ∞ to everything in it." Pretty much just saying that every portion of an infinite-dimensional space is, itself, infinite in size. That's literally all there is to it.

And this might make mathematicians rather distraught, because that makes it "Not nice," or "Not useful" to them. But since we aren't doing mathematical analysis and only care about the raw concepts, this stuff is of absolutely no importance to us. As far as we're concerned, size/measure in the physical sense does apply to infinite-dimensional space just fine. (Here's more stuff on that, for the more math-inclined)

The first claim is true (As far as I'm aware) but also ultimately irrelevant. Infinite-dimensional spaces are a tad weird compared to finite-dimensional spaces, sure, but in all the ways that matter, the two are alike.

 

[Agnaa]

That's like asking "What would you do if a verse depicted a thing that explicitly has 5 spatial dimensions as dwarfing something previously described as existing above dimensionality itself?". The answer is obviously that, if that is the case, the latter thing was just never beyond dimensionality to begin with. I'd obviously place those verses below 1-A.

I think these things are different. I think "Thing above dimensionality is bound by 5 spatial dimensions" is more clearly contradictory than "Stack of reality-fiction differences all exist within a Type IV multiverse".

And I find it a bit unsatisfactory that adding information which isn't clearly contradictory would vastly lower a verse's tier.

No such thing as a middle ground between the two, no. Either you have evidence of constituting a qualitative superiority, or you don't, and are assumed to be quantitative by default. And that's assuming superiority is even indicated at all, of course.

Those hypotheticals you bring up have issues with them, also, because they're completely unqualified statements that can be taken in a myriad ways.
For example I could take "higher level of reality" to mean levels of realness, or I could interpret "reality" as an object that has levels (In which case those levels needn't have any superiority/inferiority relation between each other at all). "new power systems where each step in it completely dwarfs the previous power system, no matter how far it's extended" is also entirely dependent on how exactly this "extension" of the previous power systems happens. I can conclude nothing from those, in short.

That is kind of the point; that they can be taken in a myriad of ways.

Still, I'd expect that if such a vague thing came up for 1-A characters, you wouldn't immediately assume that it was quantitative and bump them down below 1-A for it, am I wrong?

(Stick with me here, the punchline comes in the next post)

 

[Ultima_Reality]

I think these things are different. I think "Thing above dimensionality is bound by 5 spatial dimensions" is more clearly contradictory than "Stack of reality-fiction differences all exist within a Type IV multiverse".

And I find it a bit unsatisfactory that adding information which isn't clearly contradictory would vastly lower a verse's tier.

You asked about a hypothetical scenario where a verse "concretely establishes mathematical hierarchies as encompassing and dwarfing R>F or other qualitative hierarchies." BDE is a perfectly valid illustrative example and not at all outside of the parameters you prescribed.

And the information is indeed contradictory, yes. Those two cases are the exact same kind of thing, except the latter uses fancier terms than the former.

That is kind of the point; that they can be taken in a myriad of ways.

Still, I'd expect that if such a vague thing came up for 1-A characters, you wouldn't immediately assume that it was quantitative and bump them down below 1-A for it, am I wrong?

That question means nothing to me. You should probably rephrase it.

 

[Agnaa]

You asked about a hypothetical scenario where a verse "concretely establishes mathematical hierarchies as encompassing and dwarfing R>F or other qualitative hierarchies." BDE is a perfectly valid illustrative example and not at all outside of the parameters you prescribed.

And the information is indeed contradictory, yes. Those two cases are the exact same kind of thing, except the latter uses fancier terms than the former.

I get how you'd see both as such if you structure your standards that way, but I think an external view over all possible standards people would have, would result in many more finding the first thing I mentioned contradictory, than the second thing I mentioned.

As such, it feels strange for such verses to be nerfed by elaborating on their cosmology more. That's part of why I generally try to have stricter standards with things, so we don't end up with as many cases where saying more gets things nerfed.

That question means nothing to me. You should probably rephrase it.

There are descriptions which are obviously qualitative or quantitative.

There are also descriptions which are more ambiguous.

I believe one which is obviously qualitative would bump such a character up to 1-A, in your standards.

I believe one which is obviously quantitative would nerf that character to below 1-A, in your standards.

So, with one which is ambiguous, would you do neither. As the alternatives would be to either always assume a bump to 1-A, or always assume a limit below 1-A, or ignore them entirely (despite them being used for having verses crawl up steps through the lower end of tier 1 currently).

 

[Ultima_Reality]

I get how you'd see both as such if you structure your standards that way, but I think an external view over all possible standards people would have, would result in many more finding the first thing I mentioned contradictory, than the second thing I mentioned.

As such, it feels strange for such verses to be nerfed by elaborating on their cosmology more. That's part of why I generally try to have stricter standards with things, so we don't end up with as many cases where saying more gets things nerfed.

That honestly seems to be less a general problem and more a problem with your personal intuitions. "I think a mahlo cardinal sounds more impressive than a R>F Transcendence so it's weird that we'd rate the latter as above the former."

Stuff like this is entirely an animal reaction, no different from how very large finite numbers come off as more impressive than infinity to us, most of the time. It means absolutely nothing and your idea of introducing this as a genuine factor results in the strange idea of tiering verses based on what's effectively... points for effort. That's hilarious but ultimately absolute nonsense.

There are descriptions which are obviously qualitative or quantitative.

There are also descriptions which are more ambiguous.

I believe one which is obviously qualitative would bump such a character up to 1-A, in your standards.

I believe one which is obviously quantitative would nerf that character to below 1-A, in your standards.

So, with one which is ambiguous, would you do neither. As the alternatives would be to either always assume a bump to 1-A, or always assume a limit below 1-A, or ignore them entirely (despite them being used for having verses crawl up steps through the lower end of tier 1 currently).

Not exactly sure what you mean by "Nerf that character below 1-A" or "Assume a limit below 1-A." It seems you're trying to ask me how I'd treat statements that'd logically be 1-A at face-value, but are also just ambiguous enough for you to think I'd draw a line at them. I don't think some of the hypotheticals you've given me are 1-A even at face value, though. They're just nothing at all, to begin with.

Overall you're probably better off asking me about actual examples from verses that actually exist out there. Hypotheticals that are inherently with no context or substance hardly seem worth the time.

 

[Antvasima]

I still agree with your points, but you are being unnecessarily rude again, Ultima.

 

[Ultima_Reality]

I still agree with your points, but you are being unnecessarily rude again, Ultima.

(I'll be honest and say I didn't even realize, this time. What I'm saying doesn't register as rude to me to begin with. But, fine, I guess)

 

[Antvasima]

points for effort. That's hilarious but ultimately absolute nonsense.

I was referring to this part.

 

[Ultima_Reality]

I was referring to this part.

I suppose I'll tone that down, too. Agnaa and I are friends, so I intend this stuff as banter, not malicious jabs, but if it's inappropriate then so be it.

 

[Antvasima]

Okay. Noted.

 

[Agnaa]

That honestly seems to be less a general problem and more a problem with your personal intuitions. "I think a mahlo cardinal sounds more impressive than a R>F Transcendence so it's weird that we'd rate the latter as above the former."

Stuff like this is entirely an animal reaction, no different from how very large finite numbers come off as more impressive than infinity to us, most of the time. It means absolutely nothing and your idea of introducing this as a genuine factor results in the strange idea of tiering verses based on what's effectively... points for effort. That's hilarious but ultimately absolute nonsense.

That's not the point I'm making.

I'm saying that they're different systems, I think it's really weird to say that there's a way of comparing these different systems, that's as obviously objectively correct as "5-D can't be above all dimensions" and "Finite numbers can't be above infinite numbers", especially given the history of our site (where we didn't put R>F above those things) and battleboarding as a whole.

I'm not suggesting you should incorporate "points for effort" into your system or anything. Maybe just not giving verses lower ratings for establishing quantitative stuff on top of qualitative stuff. Or not going for this changes of systems in the first place.

Not exactly sure what you mean by "Nerf that character below 1-A" or "Assume a limit below 1-A." It seems you're trying to ask me how I'd treat statements that'd logically be 1-A at face-value, but are also just ambiguous enough for you to think I'd draw a line at them. I don't think some of the hypotheticals you've given me are 1-A even at face value, though. They're just nothing at all, to begin with.

Overall you're probably better off asking me about actual examples from verses that actually exist out there. Hypotheticals that are inherently with no context or substance hardly seem worth the time.

By "Nerf that character below 1-A" or "Assume a limit below 1-A" I mean the stuff we're talking about directly above this. How saying that an R>F difference is smaller than all of mathematics would revise how we tier the verse, such that the "R>F difference" would be treated as fake, and lowered to a tier below 1-A.

I'm not trying to ask about statements that would "logically be 1-A at face value".

I'm just trying to ask about statements that don't cleanly fall into being either obviously qualitative or obviously quantitative.

It's hard for me to provide examples on this sort of thing, because I'm only knowledgeable on literally one tier 1 verse. Hypotheticals let me get to the actual essence of something, rather than having to deal with details irrelevant to the actual question I care about. I can try to find one if you want, but I expect it would just make this discussion take way longer.

 

[DontTalkDT]

And on a less important note:


The second claim is only a half-truth. I assume you're referring to the common claim that there is no infinite-dimensional analogue of the Lebesgue Measure. For an explanation of this, I draw attention to what you said: It doesn't cleanly generalize, but it does generalize.

Basically, for those who don't know: The "Lebesgue Measure" is the standard notion of size that we use to measure solids in math. So, when we talk about the length of an object, or its area, or volume, and so on, we're grouping all that under the Lebesgue Measure. So we say "Area" is the 2-D analogue of it, "Volume" is the 3-D analogue of it, and so on.

When we jump to infinite-dimensional space, though, you often see the claim that there isn't any analogue of it there, which, as said, is only a half-truth. Math people say that because they care very much about a space having a subset of finite size (measure), since most analysis happens on the level of the finite, for obvious reasons. And so, the reason it's often said that "There is no infinite-dimensional analogue of the Lebesgue Measure" is:



In English, this translates to "The only measure that assigns finite sizes to any part of an infinite-dimensional space is the measure that says every subset of that space is of size 0. Otherwise the only available measures are ones that assign a size of ∞ to everything in it." Pretty much just saying that every portion of an infinite-dimensional space is, itself, infinite in size. That's literally all there is to it.

And this might make mathematicians rather distraught, because that makes it "Not nice," or "Not useful" to them. But since we aren't doing mathematical analysis and only care about the raw concepts, this stuff is of absolutely no importance to us. As far as we're concerned, size/measure in the physical sense does apply to infinite-dimensional space just fine. (Here's more stuff on that, for the more math-inclined)

The first claim is true (As far as I'm aware) but also ultimately irrelevant. Infinite-dimensional spaces are a tad weird compared to finite-dimensional spaces, sure, but in all the ways that matter, the two are alike.

I was not meaning to go that technical, but we can do that.
It seems you are missing the problem with what you explained. It's like saying that the 3D Hausdorff measure is a reasonable measure for 4D size, as from a 3D perspective it makes sense to claim that 4D objects are infinite.
True, it makes sense to claim that 4D objects are infinite from a 3D perspective, but that doesn't make it a sensible quantifier for 4D size.
In fact, if we go with the idea of Hausdorff dimensions, that property is exactly what shows that the 3D Hausdorff measure is a lower D measure applied to a higher D space.
This is beyond a matter of elegance. It's just not an appropriate definition of physical size at this scale.
It's like the fact that you can turn anything into a trivial topology and assign it a topological dimension of 0, it just is not really meaningful in a physical context.

The former I would not dismiss that easily either. I'm really not sure based on what you decide what matters. I would say anything that fundamentally doesn't allow the formulation of something "physics-like" is beyond what people would imagine to be physical space. This goes double if we are talking about spaces that can imbed our universe.
You might remember this debate on the smallest space that contains all finite dimensional spaces.
The space of finite sequences as you recall is incomplete, which goes severely against a layman understanding of space or "realms" so to say.
The space of all sequences meanwhile not only has no metric that reduces to the euclidean one in finite subspaces (i.e. is incompatible with our regular notion of distance), but in particular also doesn't admit to any continuous norm (if equipped with product topology). Said property is also transferred to all spaces which contain that. (In the sense that a subspace is TVS-isomorphic to R^N)
A discontinuous norm is from the perspective of physical interpretation not much better than being non-Hausforff: You see an object 1m away from you, then it moves 0.1mm to the right and is suddenly 1 million lightyears away.

 

[Ultima_Reality]

That's not the point I'm making.

I'm saying that they're different systems, I think it's really weird to say that there's a way of comparing these different systems, that's as obviously objectively correct as "5-D can't be above all dimensions" and "Finite numbers can't be above infinite numbers", especially given the history of our site (where we didn't put R>F above those things) and battleboarding as a whole.

I'm not suggesting you should incorporate "points for effort" into your system or anything. Maybe just not giving verses lower ratings for establishing quantitative stuff on top of qualitative stuff. Or not going for this changes of systems in the first place.

How we've historically treated R>F is irrelevant, seeing as I am arguing that said treatment of it is not coherent to begin with. I also find it rather odd that you talk about "qualitative stuff" as a whole here and yet direct your problems entirely at R>F Transcendence.
Especially seeing as every debate we've had over this subject in the past (Whether focused around Type IV Multiverses, or BDE, or whatever else) has, if I recall, ended with you pointing out that upgrading any of these things to their proper places means upgrading R>F as well.

Overall, though: Do you have anything to add, with this querry, outside of "I think it's weird"? If that's really all there is to it, I believe it's better to terminate this part of the conversation already.

By "Nerf that character below 1-A" or "Assume a limit below 1-A" I mean the stuff we're talking about directly above this. How saying that an R>F difference is smaller than all of mathematics would revise how we tier the verse, such that the "R>F difference" would be treated as fake, and lowered to a tier below 1-A.

I'm not trying to ask about statements that would "logically be 1-A at face value".

I'm just trying to ask about statements that don't cleanly fall into being either obviously qualitative or obviously quantitative.

It's hard for me to provide examples on this sort of thing, because I'm only knowledgeable on literally one tier 1 verse. Hypotheticals let me get to the actual essence of something, rather than having to deal with details irrelevant to the actual question I care about. I can try to find one if you want, but I expect it would just make this discussion take way longer.

I favor practical information over incredibly specific thought-experiments that likely won't ever come up (Your second hypothetical comes to mind here), so, yes, I would much rather have actual concrete examples to work with.

That said, you mentioned that all this was leading up to a punchline. So, skip the presentations and tell me what it is. That might speed up this discussion.

 

[Ultima_Reality]

I was not meaning to go that technical, but we can do that.
It seems you are missing the problem with what you explained. It's like saying that the 3D Hausdorff measure is a reasonable measure for 4D size, as from a 3D perspective it makes sense to claim that 4D objects are infinite.
True, it makes sense to claim that 4D objects are infinite from a 3D perspective, but that doesn't make it a sensible quantifier for 4D size.
In fact, if we go with the idea of Hausdorff dimensions, that property is exactly what shows that the 3D Hausdorff measure is a lower D measure applied to a higher D space.
This is beyond a matter of elegance. It's just not an appropriate definition of physical size at this scale.
It's like the fact that you can turn anything into a trivial topology and assign it a topological dimension of 0, it just is not really meaningful in a physical context.

That "3-D to 4-D" analogy is not very good because, obviously, there is no space of "infinity-minus-one" dimensions, and you can indeed define a measure directly on the space in question. So it's not like, if you define size on an infinite-dimensional space at all, you're actually only defining it based on a lower-dimensional perspective.
As you know: The principle mentioned above is basically just the fact that, if you have a ball of arbitrary positive radius in infinite-dimensional space, you can also find that it's the sum of infinite balls of smaller radii. And since these balls are all of equal volume, and adding a number to itself infinite times always gives ∞, the only way to have the first ball be finite in volume is for the smaller balls to have size 0. Otherwise it has size ∞, and so do the balls inside it.

And then, of course, you can also say each ball acting as a summand of that collection is the sum of yet another infinite collection of balls, and you can say the same for each of the balls in that collection, and so on.

Trippy, for sure, but by no means a problem for us. Nor is it "physically meaningless" as far as we're concerned. It just means everything in the space is infinite, which is a far cry from something like the trivial topology, which is just mathspeak for "Let's take some set X and pretend its members and subsets don't exist. Act like X and the empty set are the only things around."

Though the argument itself is weird, anyway. Do you think infinite-dimensional spaces have to be non-physical, or something? Given you implied that the only measures that can be had on them are ones that don't quantify physical size at all, like the probability measure (Which measures... the probability of events)

The former I would not dismiss that easily either. I'm really not sure based on what you decide what matters. I would say anything that fundamentally doesn't allow the formulation of something "physics-like" is beyond what people would imagine to be physical space. This goes double if we are talking about spaces that can imbed our universe.
The space of finite sequences as you recall is incomplete, which goes severely against a layman understanding of space or "realms" so to say.
The space of all sequences meanwhile not only has no metric that reduces to the euclidean one in finite subspaces (i.e. is incompatible with our regular notion of distance), but in particular also doesn't admit to any continuous norm (if equipped with product topology). Said property is also transferred to all spaces which contain that. (In the sense that a subspace is TVS-isomorphic to R^N)
A discontinuous norm is from the perspective of physical interpretation not much better than being non-Hausforff: You see an object 1m away from you, then it moves 0.1mm to the right and is suddenly 1 million lightyears away.

What matters is decided based on "It's a space with dimensions in it." We can chat for a while about how wacky infinite-dimensional spaces are, if you like (I don't even mean this in a deriding way, btw. Legitimately interesting topic), but ultimately it's hardly relevant if we're talking about things like "This character transcends the very quality of having dimensions at all," and things with the same result.

 

[Agnaa]

Especially seeing as every debate we've had over this subject in the past (Whether focused around Type IV Multiverses, or BDE, or whatever else) has, if I recall, ended with you pointing out that upgrading any of these things to their proper places means upgrading R>F as well.

No, that's not how those discussions ended. At some point in those discussions I did say something along the lines of "I think that's a ridiculously high-end, comparable to putting R>F differences at tier 0", but I didn't say that rating one highly entailed rating R>F highly, nor was that really the climax of the debate. In fact, I suspect it was one of the earlier points I made.

And I don't think that happened with BDE. More with omnipotence, Type IV multiverses, and apophatic theology.

That said, you mentioned that all this was leading up to a punchline. So, skip the presentations and tell me what it is. That might speed up this discussion.

Essentially, that it's strange for new hierarchies of such "vague superiorities" to lead to no meaningful progress towards 1-A from below, no matter how many of them there are. Yet for just one of them to be able to jump to High 1-A if they start at 1-A, despite them hypothetically being described the same way, just across different verses.

Our current system has something superficially similar, in that the same hierarchy-description could jump from 1-B to 1-A, or from High 1-A to 0, but that's just because the latter series already had such hierarchies earlier, and the former series could catch up by establishing more of them.

(If this sounds like it's assuming things about your position, that's because you asked me to skip right to the end.)

 

[Ultima_Reality]

No, that's not how those discussions ended. At some point in those discussions I did say something along the lines of "I think that's a ridiculously high-end, comparable to putting R>F differences at tier 0", but I didn't say that rating one highly entailed rating R>F highly, nor was that really the climax of the debate. In fact, I suspect it was one of the earlier points I made.

And I don't think that happened with BDE. More with omnipotence, Type IV multiverses, and apophatic theology.

It did, in fact, happen with BDE, and in fact it was one of the things emphasized to me back when my proposal to upgrade it to High 1-A was rejected. Though, frankly, if you think that "An uncountably infinite gap in volume" is not even an erroneous-yet-necessary compromise, or anything, but a legitimate and perfectly reasonable lowball for any of these things, then you've simply contented yourself with being objectively wrong all around, I am sorry to say. Just means the state of the arguments underpinning the current system is honestly even more precarious than I thought it was.

But I'll repeat the question I made up there: Do you have anything to add with this querry? Aside from "I think it's weird"? Or can we simply finish this part of the conversation already?

Essentially, that it's strange for new hierarchies of such "vague superiorities" to lead to no meaningful progress towards 1-A from below, no matter how many of them there are. Yet for just one of them to be able to jump to High 1-A if they start at 1-A, despite them hypothetically being described the same way, just across different verses.

Our current system has something superficially similar, in that the same hierarchy-description could jump from 1-B to 1-A, or from High 1-A to 0, but that's just because the latter series already had such hierarchies earlier, and the former series could catch up by establishing more of them.

If you don't have enough to make the jump to 1-A, then you don't have enough to make the jump to High 1-A, either, even if you start at 1-A. You don't have dimensions at 1-A, but you'd still need statements indicating you fundamentally surpass whatever it is that you got there.

Though, I think the deeper point you're trying to make here is clear: You were trying to lead me into saying that there can be quantitative differences on the level of 1-A, so you could point out a seeming contradiction with what I've said above: That if a verse establishes mathematical/quantitative hierarchies as being above supposedly qualitative layers, the latter would be deemed "fake" and downgraded to below 1-A.

That's really no contradiction, though. It's like how you can have characters with R>F Trancendence who are just regular 3-D humans in their own reality, even though they transcend everything in the lower planes, including their dimensions. That's not really the verse "establishing quantitative hierarchies that are above qualitative stuff" in the sense you meant, since a 4-D structure in the lower plane wouldn't be above the character in the higher plane. Only a 4-D structure that also belongs to the higher plane (And which thus is itself qualitatively above the lower plane) would be.

 

[Agnaa]

But I'll repeat the question I made up there: Do you have anything to add with this querry? Aside from "I think it's weird"? Or can we simply finish this part of the conversation already?

I mostly dropped this because I don't have much more to add, other than trying to publicly indicate that I don't hold beliefs you publicly assert I have.

It did, in fact, happen with BDE, and in fact it was one of the things emphasized to me back when my proposal to upgrade it to High 1-A was rejected.

Ah, my bad, I didn't process that a thread solely about AP was what you meant by a BDE thread.

Anyway, what you said occurred is demonstrably not what occurred. Three posts in I made my only mention of R>F, and I said that wasn't my biggest contention with your standards. My bigger ones were your willingness to hand out such a tier for far weaker statements than I, and you placing it at High 1-A, rather than tier 0.

So no, it wasn't the last thing I said, and no, it wasn't the main thing I emphasized, and no, I didn't say that upgrading that entailed upgrading R>F, I said that they were arbitrary but ultimately valid choices between reasonable ends, just like how R>F could be put at uncountably infinitely stronger, three dimensions stronger, or above literally anything (including other qualitative differences, mind you!)

Though, frankly, if you think that "An uncountably infinite gap in volume" is not even an erroneous-yet-necessary compromise, or anything, but a legitimate and perfectly reasonable lowball for any of these things, then you've simply contented yourself with being objectively wrong all around, I am sorry to say. Just means the state of the arguments underpinning the current system is honestly even more precarious than I thought it was.

Again, not a gap in volume, a gap in the abstract idea of power which we judge all tiers by.

And as I said in those threads, it depends on the context, it's such an uncountably infinite gap over whichever constructs they sufficiently establish as being transcended. I'd be fine with a series which only contains one timeline as having a character at 1-A for a "beyond dimensions" statement, if they didn't just toss that out, and concretely established that any amount added wouldn't change that fact.

If you don't have enough to make the jump to 1-A, then you don't have enough to make the jump to High 1-A, either, even if you start at 1-A. You don't have dimensions at 1-A, but you'd still need statements indicating you fundamentally surpass whatever it is that you got there.

So the system will change from the usual hierarchy bypassing stuff, then?

Though, I think the deeper point you're trying to make here is clear: You were trying to lead me into saying that there can be quantitative differences on the level of 1-A, so you could point out a seeming contradiction with what I've said above: That if a verse establishes mathematical/quantitative hierarchies as being above supposedly qualitative layers, the latter would be deemed "fake" and downgraded to below 1-A.

I wasn't.

It was really as simple as just:

1. Assume that hierarchy jumps that are currently valid are how you reach High 1-A.
2. Ask whether such hierarchies that aren't clearly qualitative or quantitative would count.
3. Point out the weirdness that one would make the jump, and one wouldn't, depending on where they started.

But if you're saying that new hierarchies of "higher levels of reality" or "new power systems where one step in them completely dwarfs what occurred previously" simply won't be able to reach High 1-A now from a starting point of 1-A, without further elaboration, then that's that.

 

[Ultima_Reality]

Alright, Agnaa and I sorted things out off-site. At this stage, there is nothing to be done save summarize our respective points and cast it to the votes. So, if you please: @DontTalkDT @Agnaa (And make it as short and sweet as possible)

My stance hasn't changed whatsoever, but with that said, the discussion with Agnaa had me quickly realize I ought to do a few additions to the structure of the Tiering System proposals (To an extent also inspired by @Antvasima's concerns about including different mathematical definition in the new system)

Basically, I propose we keep Low 1-A as a tier. For what? Well, basically, the collection of all cardinal infinities. Or, you could also more precisely say: The collection of all spaces of dimension κ, where κ is a placeholder for any cardinal number whatsoever (It could be 2, or 80, or 106, or aleph_2, or any other, really). In practice, you could say this is basically a tier for things that encompass all mathematical objects, so, Type IV Multiverses and the like would fall here.
To not get into too much technicalities, you can call it absolute infinity, if you like. (Though technically speaking, this is what it actually is)

Why does this tier exist? For structural reasons, mainly. Basically, since dimensionality is entirely defined in terms of mathematical objects and spaces, the collection of all such spaces does exist "beyond size and dimensionality." The collection of all dimensional spaces cannot, itself, be a dimensional space, after all, just as the set of all finite numbers cannot itself, be finite, and similar (I refer to Russell's Paradox again).
So, strictly speaking, this rating is the "first stop" for Type 2 Beyond-Dimensional Existence, tiering-wise. 1-A remains the tier for R>F Trancendences (Or, more generally, ontological superiorities), and while existing beyond dimensionality is a necessary condition for it, it's not a sufficient one. (Though, most verses with beyond-dimensional realms will accidentally define them in a way that gives them the superiority needed for 1-A, anyway, from my experience)

Now: Why is it rated below an ontological superiority? Well, because, clearly, there is no ontological distinction between the collection of all mathematical objects and sub-collections of it. It encompasses and exceeds dimensionality and size, but it nevertheless is a composite of things smaller than itself. Unlike a R>F Transcendence, where there is no such continuity between the higher plane and the lower one (Reality is not the sum of fictional things, after all).
So, it is relegated to "Low" 1-A precisely because of that: It's not quite qualitative in nature, but it's well beyond ordinary quantitative objects in a way where putting them in the same tier is plainly inappropriate (Call it "meta-quantitative," if you like)

 

[Agnaa]

With the newly announced changes, I don't think there's any inherent contradictions within the system. I just think it's not actually fixing any issues with ours (since I don't think equating R>F to being uncountably infinitely stronger is nonsensical). Plus, I think it introduces a fair bit of weirdness with indexing different verses, that make me lean slightly towards disliking it (although far more towards neutral than when I first heard of the thread).

Those concerns are:

1. As already said, I think R>F has a minimum-viable-rating of being uncountably infinitely stronger, and I don't like stepping above such things.
2. I think it's weird that a series can add information that a reasonable person could find impressive to their high-tiered characters (i.e. being equivalent to a large cardinal), yet that would end up significantly downgrading it.
3. Some things which we don't currently consider too tiering relevant gain quite a bit of importance, such as platonic concepts, and souls that are non-physical and more fundamental than the bodies they sustain. Although maybe this is a bad way of viewing it, since it's more like they're still being kept at one jump, but everything quantitative is getting nerfed significantly below them.

On top of that, I find that I generally hold stricter standards than Ultima in terms of what qualifies as actual evidence of being beyond dimensions (as in, things that can make the jump to 1-A without establishing infinitely many dimensions, under the current system). Aside from the obvious stuff of treating R>F as meeting that, it seems like this revision would loosen our standards for that significantly from how they were refined 8 months ago, and really, from how they've ever been in my time on the site. I disagree with taking the highest interpretations of those statements which can easily mean something far lower.

 

[IdiosyncraticLawyer]

Alright, Agnaa and I sorted things out off-site. At this stage, there is nothing to be done save summarize our respective points and cast it to the votes. So, if you please: @DontTalkDT @Agnaa (And make it as short and sweet as possible)

My stance hasn't changed whatsoever, but with that said, the discussion with Agnaa had me quickly realize I ought to do a few additions to the structure of the Tiering System proposals (To an extent also inspired by @Antvasima's concerns about including different mathematical definition in the new system)

Basically, I propose we keep Low 1-A as a tier. For what? Well, basically, the collection of all cardinal infinities. Or, you could also more precisely say: The collection of all spaces of dimension κ, where κ is a placeholder for any cardinal number whatsoever (It could be 2, or 80, or 106, or aleph_2, or any other, really). In practice, you could say this is basically a tier for things that encompass all mathematical objects, so, Type IV Multiverses and the like would fall here.
To not get into too much technicalities, you can call it absolute infinity, if you like. (Though technically speaking, this is what it actually is)

Why does this tier exist? For structural reasons, mainly. Basically, since dimensionality is entirely defined in terms of mathematical objects and spaces, the collection of all such spaces does exist "beyond size and dimensionality." The collection of all dimensional spaces cannot, itself, be a dimensional space, after all, just as the set of all finite numbers cannot itself, be finite, and similar (I refer to Russell's Paradox again).
So, strictly speaking, this rating is the "first stop" for Type 2 Beyond-Dimensional Existence, tiering-wise. 1-A remains the tier for R>F Trancendences (Or, more generally, ontological superiorities), and while existing beyond dimensionality is a necessary condition for it, it's not a sufficient one. (Though, most verses with beyond-dimensional realms will accidentally define them in a way that gives them the superiority needed for 1-A, anyway, from my experience)

Now: Why is it rated below an ontological superiority? Well, because, clearly, there is no ontological distinction between the collection of all mathematical objects and sub-collections of it. It encompasses and exceeds dimensionality and size, but it nevertheless is a composite of things smaller than itself. Unlike a R>F Transcendence, where there is no such continuity between the higher plane and the lower one (Reality is not the sum of fictional things, after all).
So, it is relegated to "Low" 1-A precisely because of that: It's not quite qualitative in nature, but it's well beyond ordinary quantitative objects in a way where putting them in the same tier is plainly inappropriate (Call it "meta-quantitative," if you like)

Splendid. For the record, I 100% agree with your proposed changes.

 

[IdiosyncraticLawyer]

@Antvasima

 

[Ultima_Reality]

I've already pinged Ant, so, no need for that. I'd also wait until both I and DontTalk have our own summaries out before gathering the staff members for a vote.

 

[Antvasima]

Well, I would much prefer a higher diversification and specification for our highest mathematical tiers. I thought that you stated that my concerns would be accommodated in this and a few other regards that we discussed earlier, Ultima.

If my concerns are accommodated, I obviously strongly support your revision.