The set theory I'm trying to work in right now is geared towards applying an "axiom of multifoundation" whose local maximum representation is:
The interpretation of the elementhood glyphs is that each arrow type stands for which foundation-theoretic type is "in play" for that glyph, so the up arrow signifies well-foundedness, the down arrow signifies hypersets/hyperfoundedness(?) (usually they call it something like "antifoundation," which is technically fine on the intended level), and the looping arrows refer to cofoundation (minimally, as with Quine atoms, autofoundation).
The intended "theory" is to start with an ur-element (defined as a "pure element," i.e. an element that is not a set, but which might have no "essence" besides being contained in some set), ũ, which is contained in a singleton {ũ} := 1 (in lieu of Zermelo/von Neumann ordination). The idea is that if we read the elementhood type off ũ ∈X 1, the solution to X is the up arrow, because there's nothing to a pure element's name that would automatically make of it either the head of a descending chain or a loop upon itself (since it is defined in terms of not being a set, originally). So the "layer" of this set world, consisting purely in pure elements, is ∅F, and if we start the ascent in this way, then 1F consists in sets that have elements only via the ∈WF type.
Using a loose version of (or substitute for) Quine's stratified comprehension, we then try to take a well-founded set of all sets that have elements only via the ∈WF type, which in order to not be an element of itself must then have some elements via the ∈HF type. Call this first such set 2F0. So this is a "multifounded" set, in theory. Then 3F, because it is cofounded, is not accessible by the foundation operation from the 2F-sets, but is arbitrarily greater/other than all of those (alternatively, trying to well-found the union of all 2F-sets essentially "misfires" and generates an initial 3F-set, or the only 3F-set is the universal set anyway, where the universal loop just is the way in which WF and HF mirror each other in general).
"And we are done," it looks like there wouldn't be another such thematic layer, but howso? Kant sort of joked one time about how some people thought it suspicious that so many of his important subschemes were threefold, and Arthur Eddington was sometimes described as using a little "numerological" reasoning to support an exact value of 1/137 for some important value in physics that he theorized about. So I have (for those and other reasons) come to find any finite tapering-off of a concept sequence to be questionable on its face, like why wouldn't the alternations over whichever concept just proceed onward without an absolute limit? At least when we are in the land of cosmic forces and metaphysical abstractions, maybe.
However, does epistemic graph theory then provide "semantic"(?) resources for describing other foundation-themed relations, which could exist as further elementhood types, so as to support the eventual ascent to some 4F, etc., with that well-founded world as a whole entitled something variable like XF? Why would the layering of a set world like this one, stop at just a fourth level, or indeed any finite level? (I should note that, amidst the details of this world in my other notes, the initial 2F set's cardinality is already meant to be roughly equivalent to one of the critical points of a Daedalus (not exactly Icarus) embedding.)
Are foundationalism, coherentism, infinitism, and their finitely diverse combinations, the only foundation-theoretic epistemic/logical structures/patterns, or can we derive indefinitely more structural responses to epistemic regresses than those few options, from the indefinite diversity of graph/hypergraph/etc. theory?
Revisions: "clues"
A classically-styled approach to erotetic logic and the accompanying set-theoretic semantics is to understand questions as sets of possible answers. One variant of this theme focuses on "true" answers, but as will be shown, we must be careful in describing the truthfulness of some answer-types. A prescriptive question, "Do x?" can, after all, be taken for an assertoric question of confirmation, i.e. we ask that it be confirmed whether we have been instructed to do x. (This would be the meaning of saying, "Yes," to a prescriptive question, then.) And so there would be a truth-apt expression in play, the implicit assertion of confirmation.
However, it is possible to make a question itself into a correct answer: a classical (in the "literary" sense) example is Smullyan's What Is the Name of This Book? Such answers are Quine-like, i.e. they echo the problem of Quined expressions ("Yields truth when preceded by its own quotation" yields truth when preceded by its own quotation, for example: "What is this question?" is that question, etc.). But questions can also be answers to other questions, e.g., "What is the first question in this paragraph?" is the question that entitles Smullyan's cited text.
One might "object" that the full statement of the appropriate such answers is really still an assertion, an assertion that one expression of some question is equivalent to another expression. We will waive this issue for the time being in order to explore some of the dynamics of our considerations, now.
Some questions are able to be answers to themselves, but prescriptive questions are not well-asked as such. "Do x?" is not adequately answered by, "Do x?" being repeated. Moreover, the parathetic prescriptive question, "Ask this question?" is not well-asked even in general, since asking about doing something is meant to precede doing that thing, yet here the asking and the doing are identical.
"Why ask this question?" also violates the ask-before-doing parameter, in this case inasmuch as asking why do something is meant for finding a justifier for said doings. However, it does not seem as if all why-questions must fail to be answers to either other why-questions or assertoric questions to boot. (In other words, we will assume for now that it is syntactically/logically possible for some question to have a why-question as a correct answer.)
But so there are arbitrarily many "what is..." questions that seem eminently capable of being correct answers to some/other questions. Moreover, a parathetic what-question itself can be well-asked while being its very own answer as such (again e.g. as with the title of the aforementioned book of riddles).
The trick, then, is first to show how the transfoundation types correspond to these erotetic types, if correspond they do. Offhand, it seems to me like the fact that prescriptive questions cannot be self-answers and cannot usefully refer to themselves means that the erotetic relation that they express, is equivalent to the well-foundation relation. When a what-question can usefully self-refer and self-answer is then cofoundational, whereas the entire "faculty" of why-questions is hyperfounded over the form of what-querying (e.g. infinitely asking "why" forms an infinite regress in reasoning).
Arguably, then, if we would like to speak of a zeroth-order erotetic logic, this would be the same as epistemic-imperative (pre)logic (c.f. the analysis of Åqvist/Hintikka). First-order erotetic logic is assertoric erotetic logic, or erotetic logic with assertoric answers. Questions-about-questions then serve for higher-order erotetic logic, and so on and on.
The last clue so far: go to a Moretti logic(?) for erotetic relations, i.e. a graph-thematic account of erotetic logic. Are there such relations, then, that are not "merely" one of the default multifoundational types? That a logic of questions and a logic of knowledge (in the light of epistemic regresses) should inform one another, or at least that the latter be preformed by the former, does not seem amiss. Hopefully(?), then, if extensions of the XF series are possible, some of these extensions can be read off further alternations over and distinctions in the question-answer relationship.
I have accepted J D's answer, and though it is the only one that has been offered, I would like to note that it is the "most helpful" not just by default, but perhaps because some of the considerations he raised are really the only ones that can be provided towards a well-cited kind of answer to the question. I realized after I thought of my question that the 3F level of sets wouldn't be produced in the "right" way by the excession principle applied to all the 2F sets (where those are {WF + HF}), but anyway, then, by bringing up Curry (and by implication the Curry-Howard correspondence), J D reminded me of Curry's paradox, which involves parathetic sentences; and it is those sentences (Curry's as well as the "older" ones) that play a crucial role in the characterization of the Daedalus/Icarus embeddings internal to my actual full representation of the initial 2F-set and its cardinality.
Now the Curry-Howard correspondence pertains to programming languages, and even if imperatives are not given to be processed by "logic," yet imperative programming does seem proto-logical, and so I would expect that, per my remarks about erotetic logic and parathetic questions, we should also look for something about parathetic imperatives to explain how many XF types of levels can be found. E.g., "Ask this question?" is flawed, and the flaw is of a piece with the flaw in, "Comply with this imperative" (or, worse, "Don't comply with this imperative"), so I will hope for now to refine my argument (in my notes) in light of these matters.
