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While trying to resolve the Continuum Hypothesis, I accidentally solved an entirely different, and arguably much more important, set of problems, namely the question of justifying the axioms of set theory: the ZFC axioms firstly, and arbitrarily many axioms of higher infinity besides. This is how I did so. (Btw, I happen to wonder how far Brandon Sanderson will take his settheoretic Easter eggs. There happens to be a form of set theory interwoven with graph theory, which seems relevant to the notion of Spiritwebs. So will these eggs manifest most strongly in stories/substories involving the Cognitive Realm, or is there, after all, a mathematical side to the Spiritual Realm too? And if so, will it be a form of set theory that constitutes this Spiritual aspect? Supposing it is a form of set theory, will Sanderson (obliquely or not) bring up, say, the intricate doctrines of large cardinals? (Will he bring these up even if his settheoretic Easter eggs reach their apex in the Cognitive Realm only?))
Preamble: the choice between set, type, and category theory as a foundation of mathematics
I am choosing set theory as my foundation of mathematics. It is said that category theory and type theory go together very well, in the end, even such as to say that categories are effectively reducible to types. However, in light of the historical fact that set theory won out over type theory, but has not won out over category theory, I am going to assume the following: a term refers to a set if the referent has elements; it refers to a type if the referent has tokens; and it refers to a category if the referent has elements and tokens. That being said, typology adverts more to the logical sphere, whereas elementhood is more distinctly mathematical. So a category is mathematical inasmuch as it has elements. Nothing seems to have actually been gained, then, in providing a foundation of mathematics in category theory instead of set theory.
The fundamental understanding of set theory's internal justification
In the pure theory of knowledge, there is a problem, the problem of the regress of reasoning, with four "mathematical" solutions: either the regress ends in selfjustified axioms (foundationalism), the regress forms loops (coherentism), the regress is infinite (infinitism), or the problem is unsolved (skepticism, which corresponds to J0 in justification logic). Coincidentally, the elementhood relation can be sequenced in all four of these same ways, viz. there are wellfounded sets, looping sets, infinite descending elementhood chains, and then the empty settheoretic object, that which has no elements. My fundamental claim will, then, be that wellfounded, looping, and descending sets are all justifiable modulo the positive solutions to the regress of reasoning. By implication, then, although descending sets are justifiable somehow, it is not permissible to axiomatize this justification. Justification by inference from axioms is per se nota wellfounded justification, so that only the wellfounded sets are justifiable in terms of the axiomatic method as such. And although I have a model of a justified descending set, my focus for the remainder of this discourse will be the axiomatic hierarchy. This is because it is modulo that hierarchy, that solutions to various other problems of settheoretic justification with which I am familiar, have appeared.
Justification values
Frege proposed that truth is not a predicate of an assertion, but is the reference of that assertion (if it is otherwise factually correct). This is the notion of truth values. Likewise, in my theory of set theory(!), there are justification values. Truththeoretically, the values are made to coincide with 0 and 1 on the numerical side of things, with fuzzy logic usually also having every other real number between 0 and 1 as a possible "degree of truth." There is no such bracketing required for the doctrine of justification values, and this allows us to formulate the initial axiom of infinity in a novel way, one that wears its justification on its sleeve. This is to have that axiom be, "The assertion that the initial level of infinity exists, has a justification value equivalent to that level." More concisely, have j(S) be the justification function, which takes sentential inputs S and outputs the degree of justification S has. So say: ∃S(j(S) = ω), with the very S in question being ∃ω, so that j(∃ω) = ω.
This happens to turn the entire question of justifying any axiom of infinity on its head. If every higher infinity makes possible a higher infinite degree of justification, it follows that the stronger and stronger axioms of infinity are all the more justified than the lower ones, down to the axiom of ω. Not that the initial principle is therefore unjustified: it too is infinitely adequate to the question of its own existence, of course, here.
Specific justifications of largecardinal axioms
The above might not be good enough to "explain" the justification of specific largecardinal axioms, however. Granting that this is so, I would say that we can intrinsically justify, in a Gödelian way, at least some of these axioms, not by analysis of the iterative concept of sets, but by analyzing the concept of justification itself. In other words, replace ZFC's standard background logic with a justification logic. Then you open the door (as far as I know) to at least the following axioms:
The modeltheoretic characterization: every set theory of a certain form has an initial worldly cardinal assigned to it. ZFC with justification logic is such a theory. So there is a justificationtheoretic worldly cardinal (and it is justifiable to assert that this cardinal exists).
The prooftheoretic characterization: every set theory of a certain form has a prooftheoretic ordinal assigned to it. Sometimes, to "identify" this ordinal, one has to imagine a much taller, but still countable, ordinal, that figures in what is called a "collapsing function," this function being the one through which the "identification" of the prooftheoretic number is given. Those much taller countable ordinals can be "shadows" of genuine large cardinals. ZFC with justification logic is a theory such that those shadows and their counterpart large cardinals figure in its prooftheoretic analysis. So there is an (otherwise uncharacterized) justificationtheoretic large cardinal.
The infinitarylogic characterization: some standard large cardinal axioms can be formulated in terms of infinitary logic. ZFC can be assigned an infinitary justification logic for its background. So there are largecardinal characterizations available modulo this assignment. These inherit the intrinsic justification of the logic (again), such that it is sufficiently justifiable to assert that these (they are called "weakly compact" and "strongly compact") cardinals exist. Bonus points: when you introduce strongly compact cardinals, for example, you get some other types of large cardinals below the initial strongly compact one, and you get a sizable amount of those types, too. (You don't get these with the worldly cardinals, and although it is "probable" that the prooftheoretic mirror cardinalities are much greater than the smallest modeltheoretic ones, I could tell you nothing about the interim between the mirrors and the worldlies, whereas I could at least attest to measurable and inaccessible cardinals in light of the strongly compact ones.)
From what I can tell, you can do a lot more with this justificatory template. I've "rambled" long enough for now, though, so I'll leave it to the interested reader (if there are any) to ask me about that "lot more," or to go seeking for it themselves.

I had to change my physics idea significantly due to considerations regarding infinitary logic. The idea is that the laws of physics are infinite conjunctions under an ℒ(κ,λ)structure that shifts over time, with major cosmological processes constituted by those shifts. For example, at t = 0, let the infinitary logic of a given universe be ℒ(0,0). Over the interval t(0 to 1), the structure shifts to ℒ(ω,ω), which corresponds to the initial expansion, the Big Bang. Further major shifts in expansion dynamics result from further increases in ℒ(κ,λ), so that the accelerated expansion, for instance, is a consequence of these dynamics.
The model has two grounds: an empirical observation and a major prediction. The first involves the idea that perception of a standard continuum results from existing in a dimensionality that succeeds the cardinality of that continuum, much like "complete" perception of a twodimensional structure results from existence in three dimensions. Assuming that the cardinality of a continuum is aleph1, then we assume that ℒ(κ,λ) here has aleph2 for κ (which is the variable for time's dimensionality in the system), such that we perceive time as continuous (of cardinality aleph1).
The prediction the theory makes is that at some point in the future, there will be another major shift. The equation I assigned to the shifts picks out aleph4 as the value for κ resulting from the next shift. Consequently, empirical consciousness should change accordingly, then. So if we survive to that day, we could receive predictive evidence for the model from specifics of cognitive changes at that time.

Well one prediction came true, one book earlier than I expected, another seems to have been falsified, and my theory about Adonalsium being like a city got some minor evidence behind it, haven't read DAWNSHARD to doublecheck but yeah, overall, couldn't have asked for a better RHYTHM OF WAR, except maybe he could've used commas when using the word "though," more often.

The concept of True Words
This was a concept I liked a lot in fantasy, but couldn't reconcile with the way the concept was executed. My take on it was to suppose a moral codex where each kind of good action corresponded to a letter, so that performing a sequence of good actions meant "spelling out a word" and then forming "sentences" and so on. Anyway, someone with a name in the language of good actions would have that for their True Name, and a True Word would be a word for a thing in this language.