The Dread Tiering System Revisions

[Ultima_Reality]

Yes, because you've said exactly that:

Not really what I've said, no. There is a difference between "Take an infinite-dimensional space and then try to add a single new axis to it" and "Take an infinite-dimensional space and then have a space of an even higher order encompassing it." That's why I've been speaking not in terms of dimensions per se but in terms of nested higher-dimensional realms.

 

[Agnaa]

Not really what I've said, no. There is a difference between "Take an infinite-dimensional space and then try to add a single new dimension to it" and "Take an infinite-dimensional space and then have a space of an even higher order encompassing it." That's why I've been speaking not in terms of dimensions per se but in terms of nested higher-dimensional realms.

Why would we treat realms and dimensions differently, when they're equalized to be the same at all other scales?

Why does it only jump beyond R^1 points once there's R^0 layers? 11-B through 1-B also involves R^1 points.

 

[Ultima_Reality]

Why would we treat realms and dimensions differently, when they're equalized to be the same at all other scales?

Shit gets weird in infinite dimensions. That's really all there is to it. I mean, you'd certainly think R^N (The space of all sequences) would have only countably-many dimensions, but that is just not the case.

In more practical wiki terms, this should be intuitively obvious, too. A space that's completely above an infinite-dimensional realm has to make that jump precisely because adding +1 dimension to a set of infinite dimensions does nothing. Similar to how adding +1 universe to a set of countably infinite universes does nothing, and the next biggest thing is uncountably infinite universes, instead. Hierarchical jumps get larger so as to maintain the pattern.

 

[Agnaa]

In more practical wiki terms, this should be intuitively obvious, too. A space that's completely above an infinite-dimensional realm has to make that jump precisely because adding +1 dimension to a set of infinite dimensions does nothing. Similar to how adding +1 universe to a set of countably infinite universes does nothing, and the next biggest thing is uncountably infinite universes, instead. Hierarchical jumps get larger so as to maintain the pattern.

That issue exists long beforehand; that's the point I was trying to make with the rest of my post.

And still, this does not justify treating dimensions and hierarchies of realms differently in this regard. You haven't given a reason which applies to one of those but not the other.

And I do have to wonder, would you use the same argument in the other direction? If a series had an infinite hierarchy of realms, with both the realm at the top of that hierarchy and the realm directly below that being relevant, would you rate the second-highest realm at 1-B since that's the only meaningful way to be inferior?

Shit gets weird in infinite dimensions. That's really all there is to it. I mean, you'd certainly think R^N (The space of all sequences) would have only countably-many dimensions, but that is just not the case.

I wouldn't think that; holding all possible sequences sounds like the power set. I'd expect the space of a single infinitely-long sequence to be the appropriate equivalent.

 

[Ultima_Reality]

That issue exists long beforehand; that's the point I was trying to make with the rest of my post.

And still, this does not justify treating dimensions and hierarchies of realms differently in this regard. You haven't given a reason which applies to one of those but not the other.

The example given is already justification enough, and it's been sufficiently expounded on, already. More on that down below.

I don't know what you mean with that first paragraph at all.

And I do have to wonder, would you use the same argument in the other direction? If a series had an infinite hierarchy of realms, with both the realm at the top of that hierarchy and the realm directly below that being relevant, would you rate the second-highest realm at 1-B since that's the only meaningful way to be inferior?

This hypothetical sounds like it goes:

Infinite Layers < Layer ω < Layer ω+1.

In which case, I'd rate the latter two as different levels of High 1-B+. Those layers aren't dimensions, but spaces made up of dimensions, so "Layer ω" is not the same as ω dimensions, and neither is "Layer ω+1" the same as ω+1 dimensions.

I wouldn't think that; holding all possible sequences sounds like the power set. I'd expect the space of a single infinitely-long sequence to be the appropriate equivalent.

What exactly do you mean by "a single infinitely long sequence"? A sequence of coordinates filled with 1s that goes on forever is, indeed, part of the space with uncountably infinite dimensions, and not the space with countably infinite ones, so this wording confuses me.

 

[Agnaa]

I don't know what you mean with that first paragraph at all.

You can't get meaningfully larger than aleph-one points without going to aleph-two points, yet 11-B already has aleph-one points, and aleph-two points is not reached until High 1-B+.

So why use the argument of "it has to be meaningfully larger, so it's High 1-B+" when the starting point is High 1-B, and not from any earlier structure?

This hypothetical sounds like it goes:

Infinite Layers < Layer ω < Layer ω+1.

In which case, I'd rate the latter two as different levels of High 1-B+. Those layers aren't dimensions, but spaces made up of dimensions, so "Layer ω" is not the same as ω dimensions, and neither is "Layer ω+1" the same as ω+1 dimensions.

I can see why you'd read it like that, but I was imagining the series establishing it as Layer ω and Layer ω-1.

What exactly do you mean by "a single infinitely long sequence"? A sequence of coordinates filled with 1s that goes on forever is, indeed, part of the space with uncountably infinite dimensions, and not the space with countably infinite ones, so this wording confuses me.

Oh, you're talking about "sequences" as individual co-ordinates within the space, not as in the "sequences of dimensions" that can each hold vast swaths of co-ordinates.

I find it strange that you'd get a result like this, then. When operating in the real number line, there's be aleph-one many sequences of length 1, and I don't think you'd reach a higher cardinality until you get to the power set, which would require sequences of size aleph-one. Yet you're saying that you can get something substantively larger through sequences of length aleph-zero.

What's screwing up there?

 

[Ultima_Reality]

You can't get meaningfully larger than aleph-one points without going to aleph-two points, yet 11-B already has aleph-one points, and aleph-two points is not reached until High 1-B+.

So why use the argument of "it has to be meaningfully larger, so it's High 1-B+" when the starting point is High 1-B, and not from any earlier structure?

That depends wholly on what "meaningfully larger" is said with respect to. You can't be meaningfully larger than a line by being a square in terms of cardinality, but you can be meaningfully larger than it in terms of measure.

Now, with regards to this specific case, we would expect each layer in a hypothetical hierarchy of dimensional spaces to stand in similar proportion to each other, and as such, the ω-th member of the hierarchy would have to dimensionally surpass the set of all dimensions below it, which is countably infinite (Just as any given n-dimensional space surpasses the set of all dimensions below it, which is of n-1 dimensions). Ergo, the ω-th member of the hierarchy must be uncountably infinite-dimensional.

I can see why you'd read it like that, but I was imagining the series establishing it as Layer ω and Layer ω-1.

That doesn't really make sense, since ω-1 isn't really a thing. What immediately precedes ω is the natural numbers, so there's no "highest number that comes before ω." If you have the topmost part of an infinite hierarchy, and then a layer standing immediately below it, those two layers stand as ω+1 to ω necessarily.

Oh, you're talking about "sequences" as individual co-ordinates within the space, not as in the "sequences of dimensions" that can each hold vast swaths of co-ordinates.

I find it strange that you'd get a result like this, then. When operating in the real number line, there's be aleph-one many sequences of length 1, and I don't think you'd reach a higher cardinality until you get to the power set, which would require sequences of size aleph-one. Yet you're saying that you can get something substantively larger through sequences of length aleph-zero.

What's screwing up there?

Well, not quite. The real line is composed of only a countable number of objects of length 1 (Intervals). If you stuck together uncountably infinite of those, you'd have the long line, which is a quite different beast.

For this case, what ***** it up is largely the definition of a basis; a space of a countable basis needs to have each point in it be specified by some combination of a finite number of vectors in the basis. This can happen wtih the space of all finite sequences, for obvious reasons, but not with the space of all sequences, since it has infinitely long sequences and thus subsets that can't be obtained by finite sums of vectors. Translated to practical terms: The former space can be reached by unifying all finite-dimensional spaces, whereas the latter can't.

 

[Agnaa]

That depends wholly on what "meaningfully larger" is said with respect to. You can't be meaningfully larger than a line by being a square in terms of cardinality, but you can be meaningfully larger than it in terms of measure.

Now, with regards to this specific case, we would expect each layer in a hypothetical hierarchy of dimensional spaces to stand in similar proportion to each other, and as such, the ω-th member of the hierarchy would have to dimensionally surpass the set of all dimensions below it, which is countably infinite (Just as any given n-dimensional space surpasses the set of all dimensions below it, which is of n-1 dimensions). Ergo, the ω-th member of the hierarchy must be uncountably infinite-dimensional.

Then ig my approach would go to, is it really meaningfully larger if it's another indistinct member of that infinite hierarchy? If the gap between that item, and the ones before it, isn't any different than the gap between the 981st layer and the 982nd?

That doesn't really make sense, since ω-1 isn't really a thing. What immediately precedes ω is the natural numbers, so there's no "highest number that comes before ω." If you have the topmost part of an infinite hierarchy, and then a layer standing immediately below it, those two layers stand as ω+1 to ω necessarily.

I thought that object just didn't have a proper term, rather than there being a legitimate void/discontinuity.

Well, not quite. The real line is composed of only a countable number of objects of length 1 (Intervals). If you stuck together uncountably infinite of those, you'd have the long line, which is a quite different beast.

Sorry, bad confusion of terminology. I meant "of length 1" in the sense that (1,1,1,0,0,0...) is a sequence of length 3, (1234,5678,0,0,0...) is a sequence of length 2, and (G(64),0,0,0,0...) is a sequence of length 1.

The actual size of those co-ordinates would be 0, and their distance from the origin would vary but typically just be a finite number.

For this case, what ***** it up is largely the definition of a basis; a space of a countable basis needs to have each point in it be specified by some combination of a finite number of vectors in the basis.

That's weird.

 

[Ultima_Reality]

Then ig my approach would go to, is it really meaningfully larger if it's another indistinct member of that infinite hierarchy? If the gap between that item, and the ones before it, isn't any different than the gap between the 981st layer and the 982nd?

I'd deny the premise of that argument. For there to be a similar proportion between each layer (That is, X > Y if X dimensionally transcends Y), it would have to be meaningfully larger, and so the equality of gaps that you describe can't exist to begin with. Otherwise it wouldn't be a legitimate higher layer of the hierarchy at all.

I thought that object just didn't have a proper term, rather than there being a legitimate void/discontinuity.

Yeah, nah.

That's weird.

It be like that.

 

[Agnaa]

Well, that still leaves the other stuff to be responded to.

 

[Ultima_Reality]

What are you referring to, specifically?

 

[Agnaa]

Sorry, bad confusion of terminology. I meant "of length 1" in the sense that (1,1,1,0,0,0...) is a sequence of length 3, (1234,5678,0,0,0...) is a sequence of length 2, and (G(64),0,0,0,0...) is a sequence of length 1.

The actual size of those co-ordinates would be 0, and their distance from the origin would vary but typically just be a finite number.

Practically this makes things very difficult, since literally everything that occupies or effects the entirety of that aleph-null-dimensional space would automatically convert it to an aleph-one-dimensional space. Which would upgrade every High 1-B who doesn't simply have that rating due to Composite Hierarchies (and has never effected the entirety of the stack at once).

You've clarified on the "automatically convert" part, but otherwise the point still stands; if having the co-ordinary (1,1,1,1...) be filled in a countably-infinite-dimensional space requires that space to actually be uncountably-infinite-dimensional, then most High 1-Bs would be upgraded.

 

[Ultima_Reality]

You've clarified on the "automatically convert" part, but otherwise the point still stands; if having the co-ordinary (1,1,1,1...) be filled in a countably-infinite-dimensional space requires that space to actually be uncountably-infinite-dimensional, then most High 1-Bs would be upgraded.

If that part is taken out of the way, then the rest of the point collapses as well, since the crux of the matter is precisely that (1,1,1,1...) is simply not contained in countably infinite-dimensional space to begin with. Encompassing the axes (1,0,0,0,...), (1,1,0,0,...), (1,1,1,0,...), and and so on and so forth, isn't the same as encompassing all those axes and also (1,1,1,1,1...). So most High 1-Bs won't really be upgraded, no, since you can be truly infinite-dimensional even without that, and neither would "There is an infinite-dimensional space and I destroyed it" inherently make you High 1-B+ for the same reason.

 

[Agnaa]

If that part is taken out of the way, then the rest of the point collapses as well, since the crux of the matter is precisely that (1,1,1,1...) is simply not contained in countably infinite-dimensional space to begin with. Encompassing the axes (1,0,0,0,...), (1,1,0,0,...), (1,1,1,0,...), and and so on and so forth, isn't the same as encompassing all those axes and also (1,1,1,1,1...). So most High 1-Bs won't really be upgraded, no, since you can be truly infinite-dimensional even without that, and neither would "There is an infinite-dimensional space and I destroyed it" inherently make you High 1-B+ for the same reason.

You can be truly infinite-dimensional without there being infinitely many dimensions which each contain objects?

 

[Ultima_Reality]

You can be truly infinite-dimensional without there being infinitely many dimensions which each contain objects?

You can be truly infinite-dimensional without there being an endpoint to the series of embedded higher-dimensional spaces, which is what High 1-B+ would entail there.

Also I did this:

Ultima Reality/non

vsbattles.fandom.com

Ultima Reality/imm

vsbattles.fandom.com

 

[DontTalkDT]

That whole High 1-B+ debate sounds like my cup of tea, but I can't really follow which distinction is debated in detail
Anyway, on suspicion I will point out that you can, of course, add a dimension to a countably infinite space without it becoming uncountable dimensional.

The ∞th level of a hierarchy of finite dimensional spaces, each larger than the last, I would identify with the space of finite sequences. For the ∞+1th level it would depend on what the "+1" means in detail. Just another dimension (or something equivalent)? Then it's still countable. Other differences? More debatable.

 

[Ultima_Reality]

Anyway, on suspicion I will point out that you can, of course, add a dimension to a countably infinite space without it becoming uncountable dimensional.

Never denied that, obviously. The relevant test-case for the matter is moreso "An infinite-dimensional space and then an even higher-order realm that's above and beyond it, but not above physicality/dimensionality/composition," or equivalently, "The layer of a hierarchy of dimensional spaces that comes after all the finite-dimensional ones are exhausted."

 

[DontTalkDT]

I mean, an ∞+1th dimensional space would also be "above and beyond" an ∞-dimensional space, as it is even larger (as far as one can talk about size at this level). That's why we currently require that the realm would be above in a fashion that it's a different hierarchy as a whole, not just a yet larger dimensional space. (Aside, of course, if confirmed to be uncountable dimensional in some way)
Basically, it sounds to me like the description you propose doesn't distinguish clearly enough between countable extensions of the basis and jumps to an uncountable basis.

 

[Ultima_Reality]

I mean, an ∞+1th dimensional space would also be "above and beyond" an ∞-dimensional space, as it is even larger (as far as one can talk about size at this level). That's why we currently require that the realm would be above in a fashion that it's a different hierarchy as a whole, not just a yet larger dimensional space. (Aside, of course, if confirmed to be uncountable dimensional in some way)

I don't think the "different hierarchy" business goes that hard in the new Tiering System anymore, since a "hierarchy" is really just "A set of inferior and superior levels all defined by the same basic attribute." In that sense you really only get a different hierarchy at the 1-A range now.

I'd also question whether "∞+1 dimensions" is even coherent for this case, since by doing this you haven't really added a more dimensions to the space, so there isn't many respects in which you can say it's larger. At best it's "larger" in the same way an uncountable number of infinite-D universes is larger than a single one, but this isn't exactly the difference we'd expect from one layer in a hierarchy to another, like I said here.

 

[Agnaa]

We don't consider adding an extra timeline to an infinite multiverse to make it Low 1-C.

We should not consider adding an extra dimension to an infinite stack to make it High 1-B+.

 

[Ultima_Reality]

Yeah, and I agree, hence my point was never about a single extra dimension. The crux of the matter is whether what I described above is actually properly described as "A single extra dimension."

 

[Ultima_Reality]

We don't consider adding an extra timeline to an infinite multiverse to make it Low 1-C.

We should not consider adding an extra dimension to an infinite stack to make it High 1-B+.

Side-note: Like I told you in private, I already added everything we agreed on to the sandbox pages (And also a couple other things that ended up being removed by carelessness on my part, like the "Predating spacetime isn't 1-A" section in the FAQ and etc). When can I expect you to write that list you mentioned before?

 

[Agnaa]

Yeah, and I agree, hence my point was never about a single extra dimension. The crux of the matter is whether what I described above is actually properly described as "A single extra dimension."

I see no reason why it wouldn't be, when adding a timeline would be.

Side-note: Like I told you in private, I already added everything we agreed on to the sandbox pages (And also a couple other things that ended up being removed by carelessness on my part, like the "Predating spacetime isn't 1-A" section in the FAQ and etc). When can I expect you to write that list you mentioned before?

If you're picking up my work for me, my list would just be things mentioned in previous posts of mine which were never conclusively resolved (we disagreed, you didn't care to comment, or we agreed something needed to be done but couldn't hash out the details).

But to answer your question I'm working tomorrow, and getting out of the house on Friday plus one day on the weekend. If I've sufficiently recovered, I could respond on Thursday, the weekend, or Monday.

Let's say, 25% Thursday, 40% Saturday/Sunday, 30% Monday, 4.9999% some day in the week after that, 0.0001% never.

 

[Ultima_Reality]

I see no reason why it wouldn't be, when adding a timeline would be.

Timelines and dimensional levels aren't really equivalent at all, so that's a pretty faulty comparison.

If you're picking up my work for me, my list would just be things mentioned in previous posts of mine which were never conclusively resolved (we disagreed, you didn't care to comment, or we agreed something needed to be done but couldn't hash out the details).

But to answer your question I'm working tomorrow, and getting out of the house on Friday plus one day on the weekend. If I've sufficiently recovered, I could respond on Thursday, the weekend, or Monday.

Let's say, 25% Thursday, 40% Saturday/Sunday, 30% Monday, 4.9999% some day in the week after that, 0.0001% never.

Yeah I'll pick up the work. Would rather get this done with as soon as is reasonably possible.

You all can expect a last post from me later today.