The set theory I'm trying to work in right now is geared towards applying an "axiom of multifoundation" whose local maximum representation is:

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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.

  1. 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.

  2. "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.)

  3. 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.

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    I have to admit that I do not follow your multifoundation formalism, but your link lists infinite coherentism as a fourth option. Is that a "finitely diverse combination"? I imagine there are graphs that are neither acyclic nor have an edge cycle cover, so there is more. There are even questions about the structure of infinite graphs that are undecidable in ZFC, like existence of Suslin trees. However, it strains imagination to see what this might mean for epistemic justification.
    – Conifold
    Commented Jun 17, 2023 at 11:13
  • I’ll discuss it with you via chat or email or discord if you want - gotta work thru it to understand it - it seems really interesting (I probably can’t answer it, but wanna try to understand it tho) [email protected] discord.gg/VJPfvHPf twitter.com/hmltn_2 Commented Jun 17, 2023 at 12:42
  • chat.stackexchange.com/rooms/info/146728/… Commented Jun 17, 2023 at 12:46
  • @Conifold yeah, infinite coherentism would be a 2F combo, but I don't see a way to "program" the rules I have to get to that kind of 2F from WF. And I should be open to "an unusual finite number" capping off the given sequence, I think I'm more suspicious that I ended up with 4 layers (I seem to get 4s a lot in this connection, so I just worry I'm being subconsciously numerological about it). Commented Jun 17, 2023 at 16:57
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    I was just ruminating that metaphysics is essentially a linguistic manifestation of solving decision problems in regards to establishing the logical coherence of claims, preserving locally metaphysically necessary claims, switching between domains of discourse, exhibiting semantic assent, etc. Would you be interested in speculation relating epistemic graph theory to modeling metaphysics as a way of untangling underdetermination of cyclical graphs to more hierarchical structures?
    – J D
    Commented Jun 19, 2023 at 20:58

2 Answers 2


Warning. I'm just speculating, because while I appreciate your effort to build a better formal set theory formalism, I don't have the sophistication to respond to the details of your account, and more importantly, I would argue that category theory is a more intuitive representation for modeling epistemological matters. As categories are an abstraction of sets, I wonder if what is lost in the abstraction a constraint on modeling epistemology.


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?

Here is my reasoning through canonical ideas to respond:

  1. Human beings and their epistemic theories are expressed in grammars, and the grammar most apropos to a discussion on the structural limits of theories and their taxonomies would be the Turing Machine because it captures the fundamental nature of how a grammar is used. Thus, when you present a paper with graph-theoretic descriptions of epistemological structures (where a directed edge between distinct points is logical consequence, a directed edge with the same origin and finish is a loop, and multiple directed edges of any form represent conditions, I see essentially the claim that human reason is fundamentally governed by the three primitives of a TM: sequencing, iteration, and branching. Therefore, if a graph (a categorical-theoretic structure) can be used to represent an argument (a logical structure) and be processed by a machine (a programmatic structure), we have echoes of the interrelation suggested by Lambek, Howard, and Curry, and that human epistemological thinking is a form of physical computation.

Are foundationalism, coherentism, infinitism, and their finitely diverse combinations, the only foundation-theoretic epistemic/logical structures/patterns[?]

  1. It's best to say:
  • Foundationalism is essentially the structure of dogmatic argument but is only rigidly ideological in the event that foundations are deemed incontestable, and this structure might best be visualized as a directed acyclical graph with terminal nodes.
  • That infinite regress is best understood as falibilism, because it is the nature of falibilistic epistemological thinking that no proof is ultimately secure nor is any explanation ultimately complete nor is underdetermination ever really addressed without additional data. The topological nature of this epistemology is a web.
  • The circular argument is a form of coherentism in that it looks to maximize logical consistency in the argument. The topology of this structure is ultimately and predominantly cyclical.
  1. Epistemological structures can happen in isolation with each other, and what I've experienced as epiphanic conclusions is when many disparate epistemological structures suddenly become united generally under a new epistemological structure that dovetails nicely with other epistemological structures creating a new unified structure cognitive dissonance is suddenly drastically reduced and the mind races around the structure with a flood of logical consequences suddenly available that were not available before.

  2. Two dimensions of finitude are to be considered, which are those of time and space where space is just a slice of spacetime and denotes a finite interval where logical structure can be changed with a sense epistemic modality, right? That structural changes and prohibitions are essentially expressed in possible world semantics, where as when an epistemological structure evolves over time, the constraints on possible world structures themselves change. Like in a TM, there are time and space constraints to every epistemological structure, because strictly speaking, actual epistemological discourse is in a state of evolution, where yesterday's conjectures are today's theorems, or potentially a theorem last week is a misstep when a flaw in the process comes to light. We have to separate spatial and temporal finitude from each other.

  3. Epistemological structures are composed of primitives of claims (points) and logical relations (edges) and obey simple patterns: and's, or's, etc... and those are best represented by topoi, not sets? and those claims with logic are themselves abstracted to new claims (points) that operate in "higher" topologies, or whose claims are subject to additional explication and justification in "lower" topologies that are nested in a point through specification, and as areas of epistemological space grow and are made more durable and rigorous by building and connecting webs of coherent claims.

  4. This reminds me of what I'm working on with distributed computation systems being viewed as Harel's state charts which is a very topological view of data and instructions where the fundamental problem is how to best organize data and instructions to maximize efficiency in speed, storage, communication, and compute (in our case using the actor model) in different domains of computation (say disparate subject-matter experts) modeling, maintaining, and integrating their individual workflows in various and sometimes multiple layers of abstractions.

So, after some observations, let's see if I can't say something meaningful about constraints. At the fundamental level, epistemological thinking is constrained by:

  1. It is tempting to claim that the system is constrained by primitives of strings and references to other strings which is basic Sinn and Bedeutung stuff, and that these small atomic units are constructed from an alphabet, but some of the references have to be to percepts instead, because epistemology deals with facts "out there". These relations can be modeled with topological formalisms, like graph theory or category theory.
  2. These strings and references are organized into more complex epistemological structures that are the basis of sentential logic and FOL. These relations can also be modeled with topological formalisms, like graph theory or category theory. The basic structures of proof are the Agrippan sort, and then one can consider them as finite or infinite in regards to time.
  3. That there are two levels at play in every language that uses sentential logic and FOL, the language, and the metalanguage, and that semantic ascent is a shift from presumed reality to an process of questioning reality.
  4. That set theoretic logic applies to topics that are amenable to transitivity relations only and that an ur-element is constantly introduced by a new experience, and that ur-elements are quickly integrated into the larger model of reality or grow apart from reality by replacing them with explanations which specify the ur-elements in terms of other elements.
  • The reference to Turing machines is a helpful point; I've been trying to understand Lawvere's representation of things like incompleteness and halting as examples of some "fixed point theorem," which should be applicable to the issue of parathetic sentences. And the historical evolution of knowledge pertains to what the initial 2F cardinal is supposed to be, there (I have a dependent-choice axiom whose cardinal signature changes over time and which "rides the shadow" of 𝔼, which is a WF-set of all WF-knowable sets, hence 𝔼 is itself not directly knowable per time as it changes). Commented Jun 28, 2023 at 21:48
  • As for the priority of category theory for epistemology, or the use of topoi, I've been trying to go over specific category-theory derivations to see how they work, because for now I get the impression that categories are both extensional and intensional in a way that sets are primarily just extensional. And then the topos-logic accompaniment phenomenon is something I have to square with my thought that logic has a mass-noun function on the pluralist/inclusivist level. Commented Jun 28, 2023 at 21:51
  • So your answer is a meaningful, attentive response to the question. I take it as indicating that perhaps the XF series does "run out" after a finite number of entries, which might just be the fact of the matter, no matter how wary I am of finitization. (After all, Kant did come up with a reasonable, if not incorrigible, explanation for the trichotomies of his own system.) Commented Jun 28, 2023 at 21:56
  • I don't comprehend all you say, but finitization is a metaphysically necessary end if you take sets to be constructed, as opposed to real, and physical systems of computation to be temporary as opposed to eternal which is more in line with how ontologies work as a general rule. The standard reference model is finite, the period table is finite, and the base nucleotides are finite and arguably lie at the core of explanation of just about everything in the hard sciences. Why should the formal sciences be any different? Your theory is not just mathematical, but rather is axiomatic...
    – J D
    Commented Jun 28, 2023 at 22:44
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    Let us continue this discussion in chat.
    – J D
    Commented Jun 28, 2023 at 22:52

I take J D's answer to be acceptable on both levels, but so I want to go over a reason I thought of today, to think that the XF sequence really could cap off at 3F:

  1. Imagine that there is a sequence of universal quantifiers where the zeroth-order level is for "any" (which fluctuates, in natural language, from "some" to "all"), followed by (1) "all," (2) "all other," (3) "all and only," and then (4) "all and only other."
  2. You can form a Russell set for (1), (3), and (4), but not for (2).
  3. Modulo XF, take the set of all and only other 1F-sets. This is still a 1F-set. So is the set of all and only other 2F-sets still of its elements' inner type. However, by cofoundational definition, every 3F-set is a set of itself (directly or not, i.e. as a Quine node or a Quine lattice), so they can't satisfy (4) without generating a Russell set. So the fourth universal quantifier doesn't work for 3F-sets and their upwardly open interval covers the rest of the distance of the set-theoretic universe (unto absolute infinity) without further direct variations on the theme of the 3F-type.
  4. EDIT: but there can be a stable all-set, all/other-set, and all/only-set of 3F-sets. The openness of their cardinal interval is then "caused by" the gap in the quantifier interval (since the tail of the quantifier sequence is suppressed, the openness of the 3F-interval occurs for the upwards bracket to the right).

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