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Making a small CRT to address an issue that's been bugging at me for a while. As the title suggests, this involves the upper echelons of the Tiering System, so, if you don't care for that, feel free to ignore (This message being addressed to the legions of staff members that will inevitably be pinged to this. Apologies in advance).
So, without further ado:
Basically, this is a cosmological model which posits that mathematical existence and physical existence are, in fact, one and the same, and that every possible mathematical structure exists out there as another universe, with our own universe being just one among many of those. To shamelessly quote Wikipedia here:
To give a slightly more in-depth explanation of how it works: All of mathematics is based on the idea of a formal system, which is basically a set of axioms (Basic statements that are taken as true for the purpose of an argument), coupled with an alphabet of symbols, grammar and rules which we utilize to derive results from all of the above, which, in this case, are called theorems. Set theory, the foundation of our Tiering System's higher parts, is itself defined inside of a formal system. And a Type IV Multiverse, for that matter, would basically be the collection of all formal systems.
As it stands, we currently rate this type of structure at Low 1-A to 1-A, as seen from this profile, and this thread . The logic behind it essentially revolving around the ease in which we are able to define the existence of higher-dimensional spaces: For instance, take the real number line, R, which is a 1-dimensional space. To construct a 2-dimensional space out of this, you simply need to take the Cartesian Product (i.e the multiplication) of R with itself, so, R x R would result in R^2, the 2-dimensional real coordinate space.
From there, it's not too hard to see how this can be generalized to arbitrarily large numbers (R x R x R would be R^3, 3-dimensional space, R x R x R x R would be R^4, 4-dimensional coordinate space, and so on and so forth), and as such, excluding any such spaces from the expanse of a Type IV Multiverse would be effectively the same thing as pretending that, say, the number 4 doesn't exist in the verse. At the moment we take this all to culminate into P(R), the power set of R, which is the set of all possible variations of the real numbers, which we currently equal to the cardinal aleph-2, and thus to Low 1-A, as seen in the Tiering System page:
Now, as the very existence of the thread suggests, capping this process at Low 1-A is a very, very bad practice. To see what I mean, let's look back at the real of all real numbers, R; I am sure that everyone here can agree that this set is an obscenely basic one, and something we assume exists in any verse. Now, it is likewise a very basic axiom in mathematics that, if a given set X exists, then its power set, P(X), also exists. Therefore, if R exists, its power set, P(R), also exists.
This fact works like a domino effect, which means that, if P(R) exists, then the power set of that set, P(P(R), also exists, and this process stretches into infinity. In plain english, this means that, if a verse has a cosmology where all mathematical structures exist physically, then it is not possible to restrain that scope to Low 1-A, in any way, shape, or form, because this kind of thing works entirely on the principle of "The existence of X inherently implies the existence of Y." And I should note, also, that the axiom of the power set itself is very foundational, especially for the purposes of our Tiering System: Without it, you can't even prove uncountably infinite sets exist, to begin with.
In fact, if we all of the commonly-adopted axioms of set theory, then we end up with a framework containing everything from 11-C to stupidly high levels of 1-A+. This structure is often informally known as the Universe of Sets.
And these axioms, for the matter, are just as foundational and commonly-adopted as the one mentioned above. So it seems we are arbitrarily ignoring a fairly large part of mathematics for not much of a reason, as it stands.
What I propose as a remedy for these issues, then, is: "We should allow all of the usual rules and principles of set theory to be assumed as true by default, for any verse, unless one of them is openly contradicted." Meaning that all of those things mentioned above would by default exist on the ideal level, and be able to be used for tiering should a verse specify that all mathematical systems whatsoever exist as physical ones.
I don't believe this should be too controversial a position to take: We, after all, generally assume that a verse functions the same as reality, and only disregard certain parts when something that directly contradicts it is shown.
Back to the topic itself: If we were to try and fit the Universe of Sets into the Tiering System, at first glance, it would appear to be a High 1-A structure, since, in that regard, it is a bit similar to an inaccessible cardinal: It is not a set, but rather the container of all sets, and neither is it something that can be formally referred to, or constructed, using the usual tools of mathematics. To quote the Tiering System page again:
Hold that thought in your mind, though. It'll be of importance later on.
Regardless, this kind of argument is not necessarily restricted to them alone, since the points I've made are valid in all cases, and Type IV Multiverses are really just an example of a straightforward case where they would be relevant for tiering. There are a few other cases where it would also be, such as, for instance, the notably similar concept of Modal Realism, which basically says that all possible worlds exist.
Now, don't be mistaken here, this is a very specific definition of "possible." That is, it deals strictly with logical possibility, meaning that, so long as it doesn't contradict the underlying rules of some system of logic of our choice, it is a structure that exists. For instance, if you choose to frame the set of all logically possible worlds over classical logic as a whole, then every world that does not go against the usual laws of thought (Along with two other laws that aren't too relevant here) exists, these laws being: The Law of Identity (No, not the weeb character. The assertion that, for any given thing, that thing is itself), the Law of Noncontradiction (The assertion that two opposing propositions can't both be true at once) and the Law of the Excluded Middle (The assertion that, for any given proposition, it is either true or false)
Given how basic these laws are, the range of structures that exist without contradicting them is, well, big, extends much further than even the process I outlined above. And from this I take another opportunity to stress that logical possibility is really not something that your average multiverse hinges on, and is much, much, much broader than that. For instance, in a setting that works on branching timelines, the number of alternate universes would depend on the number of states achievable in a given world, which would, in turn, also cause it to be dependent on the basic initial conditions of the universe (So, for instance, there wouldn't be an alternate timeline where the universe has more than three dimensions, or different laws of physics). All of that falls strictly under the realm of probability, and as such is much narrower than logical possibility is.
All of this is fine and dandy, of course, but why does it matter? No verse currently on the wiki functions on that kind of cosmology, yes? Might be true, but do keep in mind that I am largely outlining the consequences of taking a broader, more inclusive approach to this sort of thing, and the one that becomes more obvious following this is: If a verse affirms that all logically possible worlds are real, then it has the potential to be quite high into the system, depending on what kind of logic that refers to. Of course, case-by-case analysis applies, still.
This brings me to another possible scenario, notably more controversial than the last:
Basically, it should be noted that the concept of possible worlds isn't really about a big multiverse, or anything of the sort, and Modal Realism itself is only a very specific (And hotly debated) approach to it: At their most basic, possible worlds are just semantical tools, which come into play whenever we visualize a way in which the world could have been, with the conditions determining what is "possible" and what is "impossible" being based on chosen some set of logical laws and nothing else. So, for example, it is valid to think of a possible world where flying unicorns exist (Because the physical impossible is not the same as the logical impossible).
As explained above, this arena is stupidly broad, and although it may not necessarily exist physically and instead be just a convenient tool for philosophers to waste their time with, it is relevant for tiering one specific scenario: What if an entity is capable of actualizing any possible world? That is, if a structure, any structure, obeys the laws of classical logic, this being can bring it into existence. Of course, I am talking about (What is, practically speaking) logical omnipotence. To quote our own page on the concept:
Granted, if we say that a being's power extends over anything abiding by classical logic, then that extends way beyond set theory alone. But we can reduce that concept a bit: For example, consider a scenario where this entity can instead only actualize structures that exist by the standard axioms of set theory.
And, again, I should note: I am not exactly sure if this would impact any verse. I'm largely just laying out consequences of my initial proposals, and, so far, the only verse I know of that seems to invoke logical omnipotence as a concept is the Star Maker, where the titular character is said by the author to be able to create any conceivable world, while being itself only limited by logic, but even with the Star Maker there seems to be a fair share of caveats, like the fact its creative aspect (Which the statement refers to) is described as finite. So, yeah, this sounds controversial, but may also not have that much of a shockwave after all.
There is another case, slightly more detached from the previous ones, which probably deserves its own section:
Some then extend that even further, and posit that neither modes of thinking are enough to approach the Absolute, and so, the only way to really capture it is through your silence. This means that, indeed, trying to talk about the Absolute at all will only result in you automatically referring to something less than it. It'll always be out of the reach of your mind, so to speak.
I think a fairly textbook example of a character like this is the Swirl of the Root, from Nasuverse. As already explained in another thread. And as you can see in both, we currently treat that concept as being 1-A as a default.
Just like what I talked of in the previous section, this is quite a bad practice. Essentially because, as you've probably already gathered, 1-A exists in the same "system" as everything below it; that is to say, it abides by the same basic principles and building blocks as, say, 1-C does, and is really just far, far, far larger than it. Logically speaking, being above the reach of the language of a given system should be taken as High 1-A by default (Especially when taking the basic proposal of this thread into account), which is something already reflected in our Tiering System page. To quote it again:
And this leads us to the final part of this thread:
More specifically, I'm bringing up something called the Reflection Principle, which to put it simply, is just the fact that, if you pick any set N, and some theorem describing a property of it, you will be able to find some set containing N (Let's call it M) where this theorem also applies, and so afterwards, we say M "reflects" that theorem. For the matter, this applies in both directions (Top-to-bottom and bottom-to-top), as well
Seems fairly harmless so far, since in the usual framework of set theory, this principle is restricted to the avaliable sets, and nothing beyond, because there is nothing that can be formally talked about outside of them, and so it ultimately doesn't yield much for our purposes. However, if you take the Universe of Sets itself to be something that exists, and can be talked of like any set (Which I believe would be the case in any verse where all of math exists as something physical), then we can start talking about the properties of -that-, and so the Reflection Principle can be strengthened a lot, and allow us to derive quite a few things that normally wouldn't be possible to. To shamelessly quote Wikipedia again:
To break down what that means: Basically, the Universe of Sets, if taken as an object that can be referred to in the same way a set can, would not be the sum of collections smaller than itself, and nor would it be the power set of anything (Meaning it is unreachable through either method), and by the Reflection Principle, we are then able to show that some cardinal contained in it has those exact properties, or in other words, that an inaccessible cardinal exists. This means that the Universe of Sets would then actually be quite a lot bigger than an inaccessible, and also larger than many, many sets larger than one.
What this boils down to, is that if a verse's cosmology has all mathematical structures being manifested as things that exist, then this would lead to a Universe of Sets existing, and thereby to that form of the Reflection Principle. And, to put it bluntly, the latter would make the resulting structure 0, not High 1-A, since it'd extend quite a bit beyond inaccessibles alone.
This would doubly apply if a verse described a character as personifying Cantor's Absolute Infinity, by the way, particularly since it is defined over a kind of set theory described strictly in natural, informal language, and so when speaking of it, there is no limitation in regards to what we are allowed to consider, or refer to, because the axioms are not as so thoroughly defined and so we are not forced to play by rules as strict. In particular, one one of its principles is the axiom schema of unrestricted comprehension, which basically states that, for any property (Even ones leading to paradoxes), there is a set that has that exact property.
A thing often said of the Reflection Principle is also that it endows the Universe of Sets with a notion of ineffability, that is: If any formula describing a property true of the Universe of Sets also applies to some cardinal contained in it, how, then, can we be sure that we are really ever talking about the Universe itself, and not about a part of it? Some of you might find this familiar with the aforementioned notion of apophasis, but if you are then I'd have to stop you right there, and clarify that they are not really equivalent, only similar. The Reflection Principle does, in fact, have limits, and often hits a wall depending on what mathematical system you're working on, and this is why we say stronger and weaker versions of it exist.
As a matter of fact, apophasis itself would not be capable of being defined as any mathematical structure at all, since every object that is mathematical is naturally defined by a kind of language which we use to address it. Even treating it as a member of a larger system would, nonetheless, still be treating it as participating in a broader Universe of Sets, and therefore as abiding by the same overall logic as lower tiers, which is definitionally impossible.
As such, if we are to strive for accurate indexing, it follows that all of the concepts I've mentioned above actually land in rather advanced levels of Tier 0.
For Type IV Multiverses, it should be noted that the hypothesis itself has two versions: One posits that all mathematical structures in fact exist as physical constructes, each and every one of them being a "universe" of its own, and this version of it has really no upper bound: Anything that is mathematically coherent is included.
The other, meanwhile, is far more limited, and that is the "Computable Universe Hypothesis," which basically states that only structures defined by computable functions exist in reality. As said, this version is vastly smaller, with the most glaring reduction being the fact that it only accomodates for a countably infinite number of universes, since the set of all computable functions is also countable.
As such, a verse would actually have to specify which version of a Type IV Multiverse is being addressed in the story, since one is by no means more or less valid than the other, and as such, simply alluding to the concept without further elaboration certainly wouldn't qualify. Moreover, this is compouded by the fact that, often, Type IV Multiverses are erroneously described as being simply multiverses that accomodate for worlds with distinct laws of physics, when in fact they are immensely broader than that. Thus, if they wish to qualify for 0 (or High 1-A), the verse itself would have to specify that, in fact, all mathematical systems are encompassed in that space, and not some specific subset of that category.
And of course, there is also the matter of whether a verse will correctly depict such a structure, which could impact on the tiering severely: For example, if a space is described as containing all of mathematics and logic, and then a character or realm who transcends it is stated to reside in a finitely-numbered higher-dimensional space, then we obviously got a problem. Unless the statement that would suggest it to be a lower tier is superseded by more reliable and/or detailed descriptions, as well as additional factors like how the aforementioned space is shown to function in practice, we would defer to it.
For Modal Rsalism and possible worlds, the situation may become a bit trickier: As said before, those deal with a very, very specific definition of what it means to be "possible," that being the logically possible, which is simply anything that does not defy the basic laws of some logical system. In this case, detailed and well-defined statements are obviously preferable to vaguer, more uncertain ones: Something being stated to include "all possible worlds" or "all possibilities" would not really qualify, because not all definitions of possibility are as broad as the one mentioned above; in the context of a probability space, for example, those could be given by something as simple as a power set. In cases where evidence to adopt such a vast interpretation of a statement is scarce, we would default to the lower option.
The same process, for the matter, would apply to a being defined as capable of bringing any possible world into existence: We would need evidence to reasonably infer that this does, in fact, refer to all logical possibilities, and not to a far less robust notion of the word.
For apophatic theology, the standards would likewise need to be well-defined, because it is extremely easy to fall into the mindset that any statement of something existing "beyond comprehension" or "beyond understanding" would cause a character to qualify for one. We have to acknowledge that there is a stark contrast between simply being outside of human understanding by virtue of being something that does not interact well with our brains, and something that is outright automatically above any conceptions, or labels, or definitions that we try to impose on it.
A fairly simple example of the former would be higher-dimensional space: It is simply impossible for us to visualize what a 4-dimensional object would really look like, because our brains are by no means built to imagine this, and are instead restricted to the familiar three dimensions. Nevertheless, as has been made obvious up until this point, we can very easily define and work with higher-dimensional spaces in a mathematical context. Of course, much more mundane examples can exist: We would also not slap high-tiers whenever something is described as "beyond words" to express how shocking or overwhelming it is. Context and common sense remain key here.
As said above, though, those are simply ideas, and I expect discussion regarding the specific wording of any potential standards to occur here.
So, without further ado:
The Problem
To make a long story short, as of late, there have been a handful of characters on the wiki who have been tiered a certain way because of a certain concept that they all use: the Mathematical Universe Hypothesis, which some of you know might be vaguely familiar with, since it is also known by another name, a Type IV Multiverse.Basically, this is a cosmological model which posits that mathematical existence and physical existence are, in fact, one and the same, and that every possible mathematical structure exists out there as another universe, with our own universe being just one among many of those. To shamelessly quote Wikipedia here:
To give a slightly more in-depth explanation of how it works: All of mathematics is based on the idea of a formal system, which is basically a set of axioms (Basic statements that are taken as true for the purpose of an argument), coupled with an alphabet of symbols, grammar and rules which we utilize to derive results from all of the above, which, in this case, are called theorems. Set theory, the foundation of our Tiering System's higher parts, is itself defined inside of a formal system. And a Type IV Multiverse, for that matter, would basically be the collection of all formal systems.
As it stands, we currently rate this type of structure at Low 1-A to 1-A, as seen from this profile, and this thread . The logic behind it essentially revolving around the ease in which we are able to define the existence of higher-dimensional spaces: For instance, take the real number line, R, which is a 1-dimensional space. To construct a 2-dimensional space out of this, you simply need to take the Cartesian Product (i.e the multiplication) of R with itself, so, R x R would result in R^2, the 2-dimensional real coordinate space.
From there, it's not too hard to see how this can be generalized to arbitrarily large numbers (R x R x R would be R^3, 3-dimensional space, R x R x R x R would be R^4, 4-dimensional coordinate space, and so on and so forth), and as such, excluding any such spaces from the expanse of a Type IV Multiverse would be effectively the same thing as pretending that, say, the number 4 doesn't exist in the verse. At the moment we take this all to culminate into P(R), the power set of R, which is the set of all possible variations of the real numbers, which we currently equal to the cardinal aleph-2, and thus to Low 1-A, as seen in the Tiering System page:
Now, as the very existence of the thread suggests, capping this process at Low 1-A is a very, very bad practice. To see what I mean, let's look back at the real of all real numbers, R; I am sure that everyone here can agree that this set is an obscenely basic one, and something we assume exists in any verse. Now, it is likewise a very basic axiom in mathematics that, if a given set X exists, then its power set, P(X), also exists. Therefore, if R exists, its power set, P(R), also exists.
This fact works like a domino effect, which means that, if P(R) exists, then the power set of that set, P(P(R), also exists, and this process stretches into infinity. In plain english, this means that, if a verse has a cosmology where all mathematical structures exist physically, then it is not possible to restrain that scope to Low 1-A, in any way, shape, or form, because this kind of thing works entirely on the principle of "The existence of X inherently implies the existence of Y." And I should note, also, that the axiom of the power set itself is very foundational, especially for the purposes of our Tiering System: Without it, you can't even prove uncountably infinite sets exist, to begin with.
In fact, if we all of the commonly-adopted axioms of set theory, then we end up with a framework containing everything from 11-C to stupidly high levels of 1-A+. This structure is often informally known as the Universe of Sets.
And these axioms, for the matter, are just as foundational and commonly-adopted as the one mentioned above. So it seems we are arbitrarily ignoring a fairly large part of mathematics for not much of a reason, as it stands.
What I propose as a remedy for these issues, then, is: "We should allow all of the usual rules and principles of set theory to be assumed as true by default, for any verse, unless one of them is openly contradicted." Meaning that all of those things mentioned above would by default exist on the ideal level, and be able to be used for tiering should a verse specify that all mathematical systems whatsoever exist as physical ones.
I don't believe this should be too controversial a position to take: We, after all, generally assume that a verse functions the same as reality, and only disregard certain parts when something that directly contradicts it is shown.
Back to the topic itself: If we were to try and fit the Universe of Sets into the Tiering System, at first glance, it would appear to be a High 1-A structure, since, in that regard, it is a bit similar to an inaccessible cardinal: It is not a set, but rather the container of all sets, and neither is it something that can be formally referred to, or constructed, using the usual tools of mathematics. To quote the Tiering System page again:
Hold that thought in your mind, though. It'll be of importance later on.
Regardless, this kind of argument is not necessarily restricted to them alone, since the points I've made are valid in all cases, and Type IV Multiverses are really just an example of a straightforward case where they would be relevant for tiering. There are a few other cases where it would also be, such as, for instance, the notably similar concept of Modal Realism, which basically says that all possible worlds exist.
Now, don't be mistaken here, this is a very specific definition of "possible." That is, it deals strictly with logical possibility, meaning that, so long as it doesn't contradict the underlying rules of some system of logic of our choice, it is a structure that exists. For instance, if you choose to frame the set of all logically possible worlds over classical logic as a whole, then every world that does not go against the usual laws of thought (Along with two other laws that aren't too relevant here) exists, these laws being: The Law of Identity (No, not the weeb character. The assertion that, for any given thing, that thing is itself), the Law of Noncontradiction (The assertion that two opposing propositions can't both be true at once) and the Law of the Excluded Middle (The assertion that, for any given proposition, it is either true or false)
Given how basic these laws are, the range of structures that exist without contradicting them is, well, big, extends much further than even the process I outlined above. And from this I take another opportunity to stress that logical possibility is really not something that your average multiverse hinges on, and is much, much, much broader than that. For instance, in a setting that works on branching timelines, the number of alternate universes would depend on the number of states achievable in a given world, which would, in turn, also cause it to be dependent on the basic initial conditions of the universe (So, for instance, there wouldn't be an alternate timeline where the universe has more than three dimensions, or different laws of physics). All of that falls strictly under the realm of probability, and as such is much narrower than logical possibility is.
All of this is fine and dandy, of course, but why does it matter? No verse currently on the wiki functions on that kind of cosmology, yes? Might be true, but do keep in mind that I am largely outlining the consequences of taking a broader, more inclusive approach to this sort of thing, and the one that becomes more obvious following this is: If a verse affirms that all logically possible worlds are real, then it has the potential to be quite high into the system, depending on what kind of logic that refers to. Of course, case-by-case analysis applies, still.
This brings me to another possible scenario, notably more controversial than the last:
Basically, it should be noted that the concept of possible worlds isn't really about a big multiverse, or anything of the sort, and Modal Realism itself is only a very specific (And hotly debated) approach to it: At their most basic, possible worlds are just semantical tools, which come into play whenever we visualize a way in which the world could have been, with the conditions determining what is "possible" and what is "impossible" being based on chosen some set of logical laws and nothing else. So, for example, it is valid to think of a possible world where flying unicorns exist (Because the physical impossible is not the same as the logical impossible).
As explained above, this arena is stupidly broad, and although it may not necessarily exist physically and instead be just a convenient tool for philosophers to waste their time with, it is relevant for tiering one specific scenario: What if an entity is capable of actualizing any possible world? That is, if a structure, any structure, obeys the laws of classical logic, this being can bring it into existence. Of course, I am talking about (What is, practically speaking) logical omnipotence. To quote our own page on the concept:
Granted, if we say that a being's power extends over anything abiding by classical logic, then that extends way beyond set theory alone. But we can reduce that concept a bit: For example, consider a scenario where this entity can instead only actualize structures that exist by the standard axioms of set theory.
And, again, I should note: I am not exactly sure if this would impact any verse. I'm largely just laying out consequences of my initial proposals, and, so far, the only verse I know of that seems to invoke logical omnipotence as a concept is the Star Maker, where the titular character is said by the author to be able to create any conceivable world, while being itself only limited by logic, but even with the Star Maker there seems to be a fair share of caveats, like the fact its creative aspect (Which the statement refers to) is described as finite. So, yeah, this sounds controversial, but may also not have that much of a shockwave after all.
There is another case, slightly more detached from the previous ones, which probably deserves its own section:
The Extensions
So, currently, there are a few characters who are tiered based on a certain thing called apophatic theology (Or just apophasis, if you like), which is a school of thought that essentially posits the Absolute is completely above anything that we can conceptualize, in any way, and as such, any positive descriptions we attempt to put forth to address it would fall short of capturing what it is. Thus, in such a scenario, we can only speak of the Absolute in terms of what it is not, rather than in terms of what it isSome then extend that even further, and posit that neither modes of thinking are enough to approach the Absolute, and so, the only way to really capture it is through your silence. This means that, indeed, trying to talk about the Absolute at all will only result in you automatically referring to something less than it. It'll always be out of the reach of your mind, so to speak.
I think a fairly textbook example of a character like this is the Swirl of the Root, from Nasuverse. As already explained in another thread. And as you can see in both, we currently treat that concept as being 1-A as a default.
Just like what I talked of in the previous section, this is quite a bad practice. Essentially because, as you've probably already gathered, 1-A exists in the same "system" as everything below it; that is to say, it abides by the same basic principles and building blocks as, say, 1-C does, and is really just far, far, far larger than it. Logically speaking, being above the reach of the language of a given system should be taken as High 1-A by default (Especially when taking the basic proposal of this thread into account), which is something already reflected in our Tiering System page. To quote it again:
And this leads us to the final part of this thread:
Lowballs
As I alluded to before, placing many of the above scenarios at High 1-A is quite an immense lowball. And to explain why, I'll have to talk about the Universe of Sets again.More specifically, I'm bringing up something called the Reflection Principle, which to put it simply, is just the fact that, if you pick any set N, and some theorem describing a property of it, you will be able to find some set containing N (Let's call it M) where this theorem also applies, and so afterwards, we say M "reflects" that theorem. For the matter, this applies in both directions (Top-to-bottom and bottom-to-top), as well
Seems fairly harmless so far, since in the usual framework of set theory, this principle is restricted to the avaliable sets, and nothing beyond, because there is nothing that can be formally talked about outside of them, and so it ultimately doesn't yield much for our purposes. However, if you take the Universe of Sets itself to be something that exists, and can be talked of like any set (Which I believe would be the case in any verse where all of math exists as something physical), then we can start talking about the properties of -that-, and so the Reflection Principle can be strengthened a lot, and allow us to derive quite a few things that normally wouldn't be possible to. To shamelessly quote Wikipedia again:
To break down what that means: Basically, the Universe of Sets, if taken as an object that can be referred to in the same way a set can, would not be the sum of collections smaller than itself, and nor would it be the power set of anything (Meaning it is unreachable through either method), and by the Reflection Principle, we are then able to show that some cardinal contained in it has those exact properties, or in other words, that an inaccessible cardinal exists. This means that the Universe of Sets would then actually be quite a lot bigger than an inaccessible, and also larger than many, many sets larger than one.
What this boils down to, is that if a verse's cosmology has all mathematical structures being manifested as things that exist, then this would lead to a Universe of Sets existing, and thereby to that form of the Reflection Principle. And, to put it bluntly, the latter would make the resulting structure 0, not High 1-A, since it'd extend quite a bit beyond inaccessibles alone.
This would doubly apply if a verse described a character as personifying Cantor's Absolute Infinity, by the way, particularly since it is defined over a kind of set theory described strictly in natural, informal language, and so when speaking of it, there is no limitation in regards to what we are allowed to consider, or refer to, because the axioms are not as so thoroughly defined and so we are not forced to play by rules as strict. In particular, one one of its principles is the axiom schema of unrestricted comprehension, which basically states that, for any property (Even ones leading to paradoxes), there is a set that has that exact property.
A thing often said of the Reflection Principle is also that it endows the Universe of Sets with a notion of ineffability, that is: If any formula describing a property true of the Universe of Sets also applies to some cardinal contained in it, how, then, can we be sure that we are really ever talking about the Universe itself, and not about a part of it? Some of you might find this familiar with the aforementioned notion of apophasis, but if you are then I'd have to stop you right there, and clarify that they are not really equivalent, only similar. The Reflection Principle does, in fact, have limits, and often hits a wall depending on what mathematical system you're working on, and this is why we say stronger and weaker versions of it exist.
As a matter of fact, apophasis itself would not be capable of being defined as any mathematical structure at all, since every object that is mathematical is naturally defined by a kind of language which we use to address it. Even treating it as a member of a larger system would, nonetheless, still be treating it as participating in a broader Universe of Sets, and therefore as abiding by the same overall logic as lower tiers, which is definitionally impossible.
As such, if we are to strive for accurate indexing, it follows that all of the concepts I've mentioned above actually land in rather advanced levels of Tier 0.
Final Conclusion
Of course, for concepts belonging to such high tiers, we need standards. I expect discussion to take place regarding those, should the above be accepted, because as of now, I have only an idea of what those could be.For Type IV Multiverses, it should be noted that the hypothesis itself has two versions: One posits that all mathematical structures in fact exist as physical constructes, each and every one of them being a "universe" of its own, and this version of it has really no upper bound: Anything that is mathematically coherent is included.
The other, meanwhile, is far more limited, and that is the "Computable Universe Hypothesis," which basically states that only structures defined by computable functions exist in reality. As said, this version is vastly smaller, with the most glaring reduction being the fact that it only accomodates for a countably infinite number of universes, since the set of all computable functions is also countable.
As such, a verse would actually have to specify which version of a Type IV Multiverse is being addressed in the story, since one is by no means more or less valid than the other, and as such, simply alluding to the concept without further elaboration certainly wouldn't qualify. Moreover, this is compouded by the fact that, often, Type IV Multiverses are erroneously described as being simply multiverses that accomodate for worlds with distinct laws of physics, when in fact they are immensely broader than that. Thus, if they wish to qualify for 0 (or High 1-A), the verse itself would have to specify that, in fact, all mathematical systems are encompassed in that space, and not some specific subset of that category.
And of course, there is also the matter of whether a verse will correctly depict such a structure, which could impact on the tiering severely: For example, if a space is described as containing all of mathematics and logic, and then a character or realm who transcends it is stated to reside in a finitely-numbered higher-dimensional space, then we obviously got a problem. Unless the statement that would suggest it to be a lower tier is superseded by more reliable and/or detailed descriptions, as well as additional factors like how the aforementioned space is shown to function in practice, we would defer to it.
For Modal Rsalism and possible worlds, the situation may become a bit trickier: As said before, those deal with a very, very specific definition of what it means to be "possible," that being the logically possible, which is simply anything that does not defy the basic laws of some logical system. In this case, detailed and well-defined statements are obviously preferable to vaguer, more uncertain ones: Something being stated to include "all possible worlds" or "all possibilities" would not really qualify, because not all definitions of possibility are as broad as the one mentioned above; in the context of a probability space, for example, those could be given by something as simple as a power set. In cases where evidence to adopt such a vast interpretation of a statement is scarce, we would default to the lower option.
The same process, for the matter, would apply to a being defined as capable of bringing any possible world into existence: We would need evidence to reasonably infer that this does, in fact, refer to all logical possibilities, and not to a far less robust notion of the word.
For apophatic theology, the standards would likewise need to be well-defined, because it is extremely easy to fall into the mindset that any statement of something existing "beyond comprehension" or "beyond understanding" would cause a character to qualify for one. We have to acknowledge that there is a stark contrast between simply being outside of human understanding by virtue of being something that does not interact well with our brains, and something that is outright automatically above any conceptions, or labels, or definitions that we try to impose on it.
A fairly simple example of the former would be higher-dimensional space: It is simply impossible for us to visualize what a 4-dimensional object would really look like, because our brains are by no means built to imagine this, and are instead restricted to the familiar three dimensions. Nevertheless, as has been made obvious up until this point, we can very easily define and work with higher-dimensional spaces in a mathematical context. Of course, much more mundane examples can exist: We would also not slap high-tiers whenever something is described as "beyond words" to express how shocking or overwhelming it is. Context and common sense remain key here.
As said above, though, those are simply ideas, and I expect discussion regarding the specific wording of any potential standards to occur here.
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