Over the years of my study of set theory, I have encountered several sentences of the form *V = X*: *V = L*, *V = HOD*, *V = WF* (the exclusive assertion of the cumulative hierarchy), and (if I understand [this paper](https://arxiv.org/abs/math/0612636)) *V = HW* ("hereditarily winning").


Now, in general epistemology, there is a problem of the "regress of reasons," with three formal solutions: [foundationalism](https://plato.stanford.edu/entries/justep-foundational/), [coherentism](https://plato.stanford.edu/entries/justep-coherence/), and [infinitism](https://iep.utm.edu/inf-epis/). (There is also an "empty solution," skepticism, which corresponds to J0 in [justification logic](https://plato.stanford.edu/entries/logic-justification/#BasJusLogMat).) It occurred to me that foundationalism seems to structurally correspond to well-founded sets; coherentism seems to correspond to looping set structures like Quine atoms or, "*x* ∈ *y*, *y* ∈ *z*, *z* ∈ *x*"; and infinitism to infinite descending ∈-chains. For the purposes of this post, then, assume that ∃-sentences regarding well-founded sets are justifiable in a foundationalist way, that ∃-sentences regarding looping sets are justifiable in a coherentist way, and that ∃-sentences regarding infinite descending ∈-chains are justifiable in an infinitist way. **Implication**: axioms of antifoundation are not justified; axioms are expressions of well-founded, not antifounded, justification. This is not to say that nonwell-founded set theories are unjustified altogether, only that their epistemological origins are of a different form than that of the cumulative hierarchy.


*V = X* assertions can be understood as assertions about which ∃-sentences are true for *V*. For example, *V = L* can be interpreted as, "For all sets *x*, if *x* exists, then *x* is constructible." Let *J* be the class of all sets whose ∃-sentences are sufficiently justifiable. Is it intelligible, then, to propose something like *V = J*? In other words, "For all sets *x*, if *x* exists, then the sentence asserting the existence of *x* is sufficiently justifiable."


If we allow all three of the non-empty solutions to the justificatory regress problem, then, we can have *V* include well-founded and nonwell-founded sets, while preserving the focus (in the context of axiomatic set theory) on *WF*-sets.


Next, suppose a theory of justification values, along the lines of the theory of [truth values](https://plato.stanford.edu/entries/truth-values/). That is, let us have justification not be a predicate of sentences, but their reference. Let *S* stand for some or another sentence and have (*S*) be a function that takes sentences as inputs and outputs a number corresponding to the degree to which *S* is justified. Let 0 be the justification value for completely unjustified *S* and 1 be the justification value for completely justified *S*. Now, relax the symmetry with 0 and 1 as truth values and allow that sometimes (*S*) < 0 (antijustification) and sometimes (*S*) > 1 (hyperjustification).


Peter Koellner [says](https://plato.stanford.edu/entries/large-cardinals-determinacy/#IntHie):



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> ... when one restricts to [axiomatic] theories that “arise in nature” (that is, the kind of formal systems that one encounters in a mathematical text as opposed to examples of the type above, which are metamathematically manufactured by logicians to have the deviant properties in question) the interpretability ordering is quite simple: There are no descending chains and there are no incomparable elements. In other words, it is a well-ordering.
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He also [says](https://plato.stanford.edu/entries/large-cardinals-determinacy/#NatAxi):



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> We shall organize the statements in each interpretability degree by the relation of being more evident than and we shall speak of the collection of all statements so ordered as the *evidentness order*... In seeking axioms for theories within a given interpretability degree we seek axioms that are as low as possible in the evidentness order. Ultimately one would like to find axioms that are *minimal* in the evidentness order.
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With all these ideas in place, then...


**Two lines of reasoning in the *V = J* context**


1. *A defense of axioms of infinity*


There isn't really an argument, here, so much as a novel formulation of the initial axiom of infinity, with an eye towards an account that justifies ∃-sentences concerning arbitarily larger cardinals. Viz., let us have the initial axiom of infinity be stated as ∃*S*((*S*) = ω), with the *S* in question being the very assertion of the existence of ω itself. This is rather impredicative, granted; but anyway, besides seeming to me to be a rather "superstitious" point of view, finitism lacks the resources to speak of infinite antijustification, after all. (The surreal number -ω (more on why we are using surreal numbers, here, below) cannot be assigned by the finitist, to any *S*.) In a slogan, "Having infinite numbers makes it possible to infinitely justify some *S*," which is far as I can go in otherwise justifying the initial axiom of infinity. But note that this scheme of things stands on its head the question of justifying the larger cardinal axioms: since these make ever-higher degrees of infinite hyperjustification possible, they are more and more infinitely justifiable. So far, the only filter I've come up with on these axioms is an exclusion of the choiceless sets: [this paper by Hamkins, et. al.](https://arxiv.org/abs/1311.0814) has it that the Reinhardt, etc. sets can be formed by varying the axioms of foundation and antifoundation, so since in the occurrent system there are no justifiable axioms of antifoundation, we relegate the choiceless sets to the antifounded section of *J*.


2. *Assigning a justification value to CH*


Let forcing be an *S*-justifier, i.e. if it is possible to force some *S* to hold, then *S* is justified to some extent. I won't go into the details right now, but in the occurrent system, I came up with a modal argument for the antecedent in Shelah's result that (2ℵ*n* < ℵω) → (2ℵω < ℵω4). So there is an *S*-sequence 2ℵ0 = ℵ1, 2ℵ0 = ℵ2, etc. and since each can be forced to hold, each is justified to some extent. But modulo the modal argument and for reasons of cofinality, it is also true that 2ℵ0 ≠ ℵω. Let (2ℵ0 = ℵω) = 0, then. Using surreal numbers, we can rephrase this as (2ℵ0 = ℵω) = 0/ω. Now since any number minus itself equals zero, we can further refine this as (2ℵ0 = ℵω) = (ω - ω)/ω. Let this be the limit sentence in the relevant *S*-sequence. Abstracting over the form of the limit, then, seems to give us that (2ℵ0 = ℵ*n*) = (ω - *n*)/ω, at least if *n* does not equal zero. Presto, we have that (2ℵ0 = ℵ1) = (ω - 1)/ω, or in other words, CH is only infinitesimally unjustified.


This is about as close to a derivation of CH as we can get from this system as such, though. We also have that other Continuum hypotheses are only infinitesimally unjustified; at best, CH is minimal in the relevant order of justification. Note that there couldn't be a maximally justified maximum forcing. Now more or less by definition, the smallest strongly inaccessible cardinal, which we will here denote by *I* (with α as its initial ordinal), is not equal to the cardinality of the Continuum. So we might say that (2ℵ0 = *I*) = 0 = (α - α)/α. Then, waiving the modal argument for the antecedent in Shelah's result, we can recapitulate the relevant order of justification such that (2ℵ0 = ℵ1) = (α - 1)/α, etc. Granted, we have *S* in the α-sequence that interrupt the increase towards α; depending on your aesthetic sense of such "interruptions," you might be motivated to reject the α-picture, then, here. Again, I do have an argument in the system for restricting the powersets of the ℵ*n* overall, so for what it's worth...


**Erotetic logic and forcing**


The logical background of the above is supposed to be some form of [erotetic logic](http://awisniew.home.amu.edu.pl/dydaktyka/Unilog/2.%20Inferential%20Erotetic%20Logic.pdf). Hamkins and Loewe have given us an understanding of forcing in terms of modal logic; I wonder if it is possible to interpret forcing in erotetic terms? The only progress I've made on this front is by representing forcing as a sort of disjunction elimination: with respect to the Continuum's cardinality, for example, we have an infinite disjunctive question, where each disjunct is a possible answer to the question, and we can fiat-eliminate the disjunction by forcing an answer. Then (here's a gap in the deduction), we say something like, "The larger the forcing, the less the abstract justification value," but how to frame this in terms of erotetic justification? (The template of an erotetic justifier is describable in the system, but I won't go into the details for now.)


***Is V = J a viable option when it comes to the V = X question?*** I feel like we could apply it to the issue of extrinsic vs. intrinsic justifiers. It also seems relevant to the multiverse issue.



The justifiable universe
I find the question interesting.


But I believe that your division of set theories into justificatory types is undermined by the fact that we have instances of bi-interpretable theories that cross the type boundaries.


For example, ZFC set theory is bi-interpretable with the antifoundational theory ZFC-foundation+AFA. Basically, inside ZFC, one can model anti-foundational set theory by considering the class of pointed extensional digraphs (up to isomorphism), and then proving that AFA holds for this interpretation. Conversely, in ZFC-foundation+AFA, one can consider the well-founded part of the model, which will satisfy ZFC. This is a bi-interpretation, because the AFA model can observe how it itself arises as the interpretation by means of extensional digraphs built using only vertices and edge relations in the well-founded part of the model.


The usual attitude we have toward bi-interpretable theories is that they have essentially the same content. And therefore I would find them also to be justified to the same extent by any form of justification.


The bi-interpretation extends to all extensions of these theories, such as ZFC + large cardinals, which will be bi-interpretable with a corresponding AFA theory.


Meanwhile, your taxonomy would seem to place the ZFC set theories in the foundationalist justificatory realm and the anti-foundational theories in the coherentist realm. So it seems that we should be equally able to support either theory with either form of justification, and I take this to undermine the question. After all, if we can provide a foundationalist justification of theory T, which is bi-interpretable with theory $T'$, then this same justification also seems to provide justification for $T'$, even when you would seek only coherentist justifications for $T'$.


