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Tiering System Revisions - Part 3

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@MasterOfArda

Of course. I specified in the first thread that The New System assumes the Continuum Hypothesis (specifically it's generalized form) holds, for simplicity's sake, although I probably should have clarified this asusmption in its continuations as well.
 
That is ridiculous. You can't just go around assuming whatever you want to make your system nice. Isn't it supposed to match with mathmetics? That's what this whole thread is about. All that talk about worldly cardinals and inaccessible cardinals; we weren't using hubdub cardinals because we were trying to make something that matches with mathmetics, even if declaring a new kind of cardinal (The hubdub cardinals! The smallest cardinals whose tier is directly above blahblahblah) would make it easier. You can't just go around declaring whatever you want; so much time and effort looking to match the tiering system with actual mathmetics and then that!

I know CH is not known to be false, but it is not known to be true either. I know non-staff aren't supposed to be debating, but that is so insane I had to object. We spent so much time trying to make our dimensions match with the standard conception of dimensions, but just ignore that when we get to anything above?
 
I don't really understand why this is so ridiculous in your view. The Continuum Hypothesis is not known to be either true or false in the context of a given Set-Theoretical Universe, yes, I am aware of that, but one can still include it as an additional axiom alongside whatever theory you are using, and there wouldn't really be a problem with that. As you probably already know, both the assertion that the CH is false and the assertion that it is true are consistent with the axioms of ZFC anyways.
 
That is not true. There are many mathemitcal theorems which are not directly about the observable universe but are definetly true.
 
Ngl, anything that is above High 3-A would not make any sense according to our laws of physics and such, and even that tier in particular is kinda sketchy.
 
That is ridiculous. You can't just go around assuming whatever you want to make your system nice. Isn't it supposed to match with mathmetics?

That's what mathematics does already. ZFC is built out of many axioms, adding an axiom so that the GCH holds isn't an issue, and is something many mathematicians do already.

I know CH is not known to be false, but it is not known to be true either.

I thought it was more than that - that we know that the CH cannot be proven false or true only using the axioms of ZFC, and a coherent theory can be made either by assuming it's true or assuming it's false.

@Everyone else I'd rather this thread doesn't devolve into "We can't know anything about the higher tiers anyway", I think this point can be made easily enough without going to those arguments.
 
The ZF axioms are used because they have huge extrinsic justification for well-founded sets. They are used because they are so likely to be true, it is a short distance from doubting them to Cartesian skepticisim. I can send you articles explaining this (Including, I think, by Zermelo himself).

On the other hand, GCH is not obviously true, and there are good reasons to doubt it. Talking from a Platonist perspective, I am not saying it is false, and I think because of developments in inner model theory and universally Baire sets, CH is likely true, but my opinion doesn't matter. I can think whatever I want, but it just change the fact that for numerous reasons, it is unreasonable to doubt the ZF axiom but it is reasonable to doubt CH.

Bassically, my objection is that the point of the wiki is to find out how actually strong characters: If you plopped them into reality, how strong would they be? Assuming CH calls on to much speculation. It would amount to adding a "Possibly" in front of every characters tier: They are this tier, if CH holds, then they are this tier. It says little about how strong they are if CH fails.
 
I strongly object to calling any axioms "likely to be true", since "truth" can only be established in reference to other axioms.

We have to either arbitrarily choose that CH holds or that it fails, both of which have equal backing within ZFC, and since assuming it holds seems more common in talking about infinities, and since assuming it holds makes the system much simpler, I'm fine with assuming it holds. If you have a good alternative I'd love to hear it.

We need to make a lot of assumptions when tiering characters regardless, many are much more egregious than saying something which cannot be proven true or false is true for simplicity.
 
I strongly object to calling any axioms "likely to be true", since "truth" can only be established in reference to other axioms.

That's why I said likely to be true. The only justification for them are heuristics, but they are good heuristics. My alternative is using R^¤ë(1) instead of R^R. I think it eliminates every problem and is not that complicated.
 
R^R is more reasonable as it would indicates aleph to the power of another aleph. Rather than R with the power of omega.
 
Can you explain what R^¤ë(1) means? I get what R^¤ë means.

Then why not R^ÎÉ(1 (Nothing I can do about the formatting)

I'm fine with R ^ Aleph one, but I'd like others to weigh in on this.
 
And what is ¤ë(1)?
 
On the other hand, GCH is not obviously true, and there are good reasons to doubt it. Talking from a Platonist perspective, I am not saying it is false, and I think because of developments in inner model theory and universally Baire sets, CH is likely true, but my opinion doesn't matter. I can think whatever I want, but it just change the fact that for numerous reasons, it is unreasonable to doubt the ZF axiom but it is reasonable to doubt CH.

It indeed isn't, but we are working in the framework of a Universe wherein the Continuum Hypothesis holds in the first place, speaking strictly of the proposed new system, which is perfectly consistent and not at all a problem, as I said. Anything else is irrelevant.

Bassically, my objection is that the point of the wiki is to find out how actually strong characters: If you plopped them into reality, how strong would they be? Assuming CH calls on to much speculation. It would amount to adding a "Possibly" in front of every characters tier: They are this tier, if CH holds, then they are this tier. It says little about how strong they are if CH fails.

It is not, though. We already make some heavily mechanical assumptions in the context of the wiki that most certainly wouldn't fly on our own physical universe; The purpose of this wiki is to quantify characters in the context of their own fictional setting, and that's really all there is to it.

Whether the CH holds or fails is sorta irrelevant to the tier of a character, too, since Low 1-A is primarily supposed to be an arbitrary metric space whose basis has cardinality equal to c. I already said it is assumed to be true in order to ensure the system is simple and straightfoward, and the metric turns out to be something solid and not overly obscure. If you assume the CH fails then the cardinality of the continuum could be literally anything below aleph-omega, which is obviously not intuitive and makes things unecessarily fluid.
 
¤ë(1) is the same aleph-1. They seperatly when used as the cardinality of a set, or the order-type of a well-ordered set.

We already make some heavily mechanical assumptions in the context of the wiki that most certainly wouldn't fly on our own physical universe.

What mechanical assumptions? We make some assumptions about the existence of certain objects, but as few as we can. Basically, we don't nedd to assume CH. It doesn't even make things that much simpler. Maybe a paragraph to explain ¤ë(1). It is a variant of Occam's razor: We can have an equally coherent, consistent, and simple system with one less assumption.
 
@Ultima Thanks, does ¤ë1 not solve the issue of the cardinality of the continuum possibly being anything below aleph-omega?

@MasterOfArda Why is ¤ë1 more useful than R? What problems does it avoid and why?
 
It is more useful because it avoids the need for the additional assumption ¤ë(1)=|R|. ¤ë(1) is basically a substitute c, that removes the need for the assumption of CH. Also, I think c can be any infinite cardinal of uncountable cofinality, not just does below aleph-omega.
 
If it is that simple it should be fine to use, especially since this complicated stuff would be buried in an explanation page and left out of the Tiering System page anyway.

Thoughts? @Ultima
 
What mechanical assumptions? We make some assumptions about the existence of certain objects, but as few as we can.

Assumptions such as the fact a given character can use their abilities outside of the scope of their verse, in cases where that would normally be a massive drawback, for example. Psykers from Warhammer 40k wouldn't be able to use their abilities in a neutral setting where the Warp doesn't exist, so we just assume a given battle between such character and another take place in a mixed verse where their power source holds. This is especially proeminent in battles between characters whose verses have functionally incompatible cosmologies.

The point is that this wiki is not actually about measuring how strong a character would be if they actually existed, but rather about how strong they are in a fictional context, which makes that point effectively moot, especially when fiction defies real life principles all the time.

It is more useful because it avoids the need for the additional assumption ¤ë(1)=|R|. ¤ë(1) is basically a substitute c, that removes the need for the assumption of CH.

My point is that I don't really see why assuming CH holds is a problem. Your comments imply that it goes against the notion of being strictly rigorous to Mathematics, when it really doesn't. To reiterate my previous comment, Low 1-A is just supposed to be an arbitrary metric space with basis c in the first place, and we assume CH in this case because it makes the system more straightforward and simpler to correlate with the hierarchy of Aleph Numbers in the first place. Although I guess that in such case we can just use Beth Numbers as the primary thing >~>

Also, I think c can be any infinite cardinal of uncountable cofinality, not just does below aleph-omega.

That's right, I forgot about the fact that c cannot equal aleph-omega because of the latter's nature as a singular cardinal with cofinality ¤ë, rather than its size. My bad, although that just reinforces my point.
 
I think the only thing we disagree with is CH. The problem with assuming CH is simply that is an additional assumption that is not needed. We are basically saying that a character has a certain tier if CH holds. I object to adding any assumptions that are not needed. I adds an unnesccary level of uncertainty. The less axioms we can get away with the bettter; removing CH is one less axiom with no negative consquences.
 
I'll unfollow this thread since there seems to be much more mathematical mumbo jumbo to come and I don't feel like going through all of such things. If any important development occurs here just leave a message on my message wall (along with some context behind it).
 
@MasterOfArda

Thank you for trying to help us out.
 
A second problem has just occured to me. You can have spaces of dimension ¤ë+1, ¤ë+¤ë, etc. There dimension is countable infinite, but >¤ë. You can't even Continuum Hypothesis them away because there existence is a theorem of ZF(C). Just take the set of ¤ë+1-tuples of reals, to get a space of dimension ¤ë+1, even though ¤ë+1 is not a cardinal.

I have a book on topology that I'll grab and see if it has any reference to ordinal spaces. Maybe that will help?

But anyway, let's first finish up the CH discussion while I look through the book.

Edit: Maybe existing in dimensions between ¤ë and ¤ë(1) could be considered a really strong hax ability?
 
And here I thought we were finally making some progress, but now we have to return to arguing about...whatever this is about. I should've known better.
 
We are basically saying that a character has a certain tier if CH holds.

We are not. I already said Low 1-A is just equivalent to a space with a dimension equal to the real numbers (For reasons more aligned to Topology, admittedly), and assuming GCH is just a way to streamline the process of making different subtiers and have an exact answer as to where they are positioned in the hierarchy. If you assume CH fails then everything just turns unecessarily fluid and we are back to stage zero.

Really, this is just being needlessly picky. No offense.

You can have spaces of dimension ¤ë+1, ¤ë+¤ë, etc. There dimension is countable infinite, but >¤ë. You can't even Continuum Hypothesis them away because there existence is a theorem of ZF(C). Just take the set of ¤ë+1-tuples of reals, to get a space of dimension ¤ë+1, even though ¤ë+1 is not a cardinal.

That would simply be a higher degree of High 1-B, though. Spaces with uncountably infinite dimension are so far removed from those with countably infinite dimension that they may as well be in another tier entirety, and assuming Low 1-A equals what I previously mentioned, they act as the bridge between High Hyperversal and Baseline Outerversal.

So, yeah, you could have a space of dimension corresponding to any countably infinite ordinal number, and that would still be just varying degrees of High 1-B.
 
So Low 1-A is any character of continuum many dimensions or higher, and High 1-B is any character with less then continuum many dimensions?

Assume we have a verse where one character is stated to have dimension ¤ë(1) , another has dimension c, and the second character is stated to be infinetly higher dimensions then the second. How would we resolve that?

Would the character of dimension ¤ë(1) be High 1-B because he has less then c many dimensions, or Low 1-A because he has more than countably many dimensions. Replacing c with ¤ë(1) solves that problem.
 
After rereading the OP, it seems that c was chosen because it is believed to give some transcendence over the concept of dimensions. It does not, and I will make a post explaining that in a little while, so I can give details why it doesn't work.
 
Reading through the posts above, I wouldn't mind doing things this way:

  • R¤ë up to R¤ë1 = High 1-B
  • R¤ë1 up to R¤ë2 = Low 1-A
  • R¤ë2 and anything above that = 1-A
  • A strongly inaccessible cardinal and beyond = High 1-A
At the very least, we should clarify that (for example) when we say RR, we really mean R for any ordinal n that has a cardinality of ÔäÁ1.
 
@MasterOfArda

Assume we have a verse where one character is stated to have dimension ¤ë(1) , another has dimension c, and the second character is stated to be infinetly higher dimensions then the second. How would we resolve that?

I don't think there is a need to worry about that, as most verses don't get neck-deep into the specifics of the metric, and at most drop a few statements regarding dimensions being uncountably infinite in number, in which case we just equal to the Continuum for simplicity's sake, and if we don't have further context regarding the statement.

Hypothetically speaking, though, I believe we would just consider such characters as either unquantifiable or arbitrarily high into 1-A, for lack of a better choice. It's something we wouldn't really be able to avoid even if we assumed CH is false.

After rereading the OP, it seems that c was chosen because it is believed to give some transcendence over the concept of dimensions. It does not, and I will make a post explaining that in a little while, so I can give details why it doesn't work.

Before people start saying that the following is too complicated for this wiki, I should note that I agree with that, and that even this mumbo jumbo is going to only barely be present in any separate explanation page that is to come. So, don't panic pls, it's just hidden clockwork which I am bringing up solely at this moment for the sake of giving a full explanation. k thnx

Not in the sense that it allows you to "tRanscEnd tHe cAwnCepT of dImenShuNs", no. The Real Numbers were choosen as the line between High Hyperversal and Baseline Outerversal because of their relation to Topological Manifolds and the concept of a metric, mostly.

As you can gather from previous threads and stray commentaries present through this one, the new system equates the size of any given higher-order space to continuous multiplications on R (real coordinate spaces R ^ n, pretty much). Real Coordinate Spaces are themselves topological manifolds and are required to be second-countable, hausdorff spaces by default. If one disregards those (equivalent) conditions entirely, then they cease to be metrizable (meaning they are no longer homeomorphic to a metric space) and to admit a given metric embedded in their structure. This obviously complicates things quite a bit and enters the domain of spaces which are neither hausdorff nor metrizable (and which by extension do not admit a measure, either), and thus raises the question of "what kind of metric do you use in such cases"?

This relates to spaces of dimensionality equal to the real numbers because of the fact that a second-countable space can only have at most continuum cardinality. Such spaces are not possible past this specific point, and thus the very usage of topological manifolds such as the aforementioned real coordinate spaces becomes sorta moot, as even an infinite-dimensional space (R ^ N, for example) would still have cardinality equal to R and fall under the category of stuff below Outerversal.
 
I am not that well versed in topology, so correct me if I'm wrong, but wouldn't R¤ë1 be second uncountable? If so, wouldn't that basically mean that ¤ë(1) is doing the same job as c but without the additional assumption of CH?

If that's not the case, then we can use R^R, but if it is, I think we should use R¤ë1. As for the other comment on inaccessible cardinals, what does it even mena for High 1-A to be inaccessible?
 
Sera EX said:
And here I thought we were finally making some progress, but now we have to return to arguing about...whatever this is about. I should've known better.
Click to expand...
It is best if we try to proceed, yes.
 
MasterOfArda said:
I am not that well versed in topology, so correct me if I'm wrong, but wouldn't R¤ë1 be second uncountable? If so, wouldn't that basically mean that ¤ë(1) is doing the same job as c but without the additional assumption of CH?
If that's not the case, then we can use R^R, but if it is, I think we should use R¤ë1. As for the other comment on inaccessible cardinals, what does it even mena for High 1-A to be inaccessible?
Click to expand...
high 1-A is basically strong limit K or in better terms strongly inaccessible. aleph 1 to aleph omega is weakly incaccessible. being strongly inaccessible is beyond the scale of it even fi u have uncountably infinite number of alephs.
 
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