I've been engaging in a few discussions lately about infinity and why you can't tier characters based on "higher" infinities. Allow me to explain something about infinity. Infinity does not work the way finite numbers work.
Let's say you have infinite 1D line. That line is PROVABLY and objectively the exact same size as an infinite 2D plane. The proof is pretty simple.
First off, two sets are defined as the same in size if their cardinality are the same, or in other words if you can match up their elements in a one to one fashion. I can fit every point in an infinite 2D plane into a infinite 1D plane Take every point on the 2D plane and give them an x and y coordinate. Then put them at point (2^x * 3*y). Every number has a prime factorization and therefore every number will have a unique point on the 1D line. You have shown that you can fit every point of the infinite plane on an infinite line. You can do this with 3 dimensions as well if you want. Give coordinates x, y, and z to every point to the 3D space, and assign them to point (2^x * 3^y * 5^z). Again they will each have a unique position. Because there are infinite prime numbers you can do this with literally any number of dimensions including infinite.
Allow me to intercept the expected argument. This also works for Aleph-1 (the lowest uncountable infinity) which, if you believe continuum hypothesis (and the math works out fine either way; the continuum hypothesis has been shown to be neither provable nor disprovable) is the infinity of the continuum. Raising Aleph-1 to a power does not raise it's cardinality. If you're saying that an infinite 1D line really should be Aleph-1 because "continuum" the math works the exact same. The product of two infinite cardinals is just the larger of the two.
Dimension Tiering works on the assumption that adding a dimension will somehow raise the cardinality by an uncountable infinite amount which is simply objectively untrue. There ARE larger infinities, however they can not describe physical amounts. You get a larger cardinality of infinity by taking the powerset of the prior cardinal of infinity, which is the number of possible subsets or combinations of the number. Aleph-1 is the number of subsets in Aleph-0. Aleph-2 is the number of subsets of Aleph-1 etc.
The other argument you hear sometimes is "it's fiction, it doesn't have to work by the way reality works". See that applies to most things. In fiction we have to accept what the author says if we're using their hypothetical reality....but only in the circumstance that what they are saying is actually coherent. If the author writes down a string of arbitrary characters we have full right to simply ignore it as meaningless. This is what illogical things like claiming claiming that a character is physically stronger then infinity is.
If you accept that in real life, that Aleph-1 times Aleph-1 is just Aleph-1, then let us analogize this for a second. Imagine a character A that can produce 10 gigatons of force explicitly and a second character that can also produce 10 gigatons of force explicitly. Now imagine that the series insists Character A produces more force then Character B despite them both at that time producing explicitly 10 gigatons. It's clear self-contradiction. This is because the terms "equal to" and "greater then" are mutually exclusive. Saying a character can produce MORE then infinite physical energy is equally self contradictory. Yes, we allow for irrational things in fiction all the time, but only to the extent that they actually convey meaning. Something that is meaningless, that is nonsense does not spontatenously gain coherence because we accept a hypothetical reality as a form of reality for a time. To quote C. S. Lewis:
“His Omnipotence means power to do all that is intrinsically possible, not to do the intrinsically impossible. You may attribute miracles to Him, but not nonsense. This is no limit to His power. If you choose to say, ‘God can give a creature free will and at the same time withhold free will from it,’ you have not succeeded in saying anything about God: meaningless combinations of words do not suddenly acquire meaning simply because we prefix to them the two other words, 'God can.' It remains true that all things are possible with God: the intrinsic impossibilities are not things but nonentities. It is no more possible for God than for the weakest of His creatures to carry out both of two mutually exclusive alternatives; not because His power meets an obstacle, but because nonsense remains nonsense even when we talk it about God.”
Does that mean if a character destroys infinite physical space that they are as strong as they possibly could be? No. Physically Infinite Power is only Tier 2 in my tiering system afterall. This is only for how strong they PHYSICALLY are. If you are a materialist, that's all there is, but if you accept the immaterial, as most fiction does, then there can be stronger characters. They just aren't "physically" stronger, per se.
Reality can be seen as a spectrum of more real to less real, much like a color spectrum. Fiction generally divides this spectrum by "planes" of reality. Any division of this spectrum will ultimately be arbitrary unfortunately. Just as division of the color spectrum into 7 colors, ignoring non-visible light spectrums that are at wavelengths too extreme too either side, so too is dividing reality into 7 planes ultimately arbitrary. Still this allows for potentially infinite levels of power ranking from highly un-fundamental to absolutely fundamental.
I personally call the maximally real characters in fiction "absolute plane" beings just because the term absolute means the maximal possible value. Other terms might be existence in itself, being, etc. Metafictional Manipulation is a form of absolute manipulation. The reason for this, to be rather tautological, is that to fictional beings, "being" is fiction. This brings up a related point.
Omnipotence. Obviously no fictional character is actually omnipotent. Anyone who thinks a fictional character ACTUALLY has the ability to destroy the planet is likely insane. What people who claim a fictional character is omnipotent mean is that the character is hypothetically omnipotent. The problem is that any character that is in a hypothetical is inherently bounded to fiction and so could hypothetically be attacked by a metafictional attack since the hypothetical omnipotent is just that. "Hypothetical", as in a fiction.
What are the strongest characters in fiction like? I've seen people actually try to use quantification on such characters despite the fact that the strongest characters are widely recognized as transcending concepts; including concepts like space, time, speed, power, strength, infinity, and quantification. As such any quantification will obviously not apply.
It would seem to me that the strongest characters in fiction, what I call Tier 1 characters, are characters that are absolute plane beings, absolute plane manipulations who can destroy the entire range of concepts and forms.
Obviously this is so esoteric that any human mind can not really picture it. If you try to imagine the absolute, you automatically assign concepts to it like number, color, shape etc. All of these are concepts hypothetically transcended by a absolute being.
Still, in our fictions we do have absolute beings, beings that are hypothetically transcendent of concepts despite our imperfect ability to express them. Such beings would be in theory, expressions of pure being without any form or concept, meaning there are no patterns or recognizable concepts that apply to them.
It's an odd observation that with the strongest characters, the tier 1s, you are no longer really observing what they can do but instead what they can't do. As above, any fictional character is bounded by the very fact that they are fictional and so can be manipulated hypothetically by metafictional manipulation, but the question with Tier 1s is what exactly are the things that bind them.
Anyway, hope that helps.
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