Caveat: this question is fairly technical in nature, and I have reason to believe it would be more fitting for the MathOverflow, especially in terms of potentially informative responses (there are some here who might be able to "put me in my place" but there are more who could do the same, over there). However, I don't know much mathematics; even my grasp of set theory is heavy on the natural-language/philosophical side of the topic, and although after years of dying inside while trying to read through various demonstrations, I have picked up a few tricks, still, I am hardly able to contribute to even the MathSE, much less the Overflow, in such good faith. Moreover, then, since the following will in fact be relatively light on formalism, and philosophically geared to boot, I will submit this question here, pending recommendation that it be transferred elsewhere. (So this might be one of those "questions that occur in a philosophical context," without somehow counting as philosophy. I'll leave it to the more established/informed contributors, here, to decide.)
The first-order problem. So there is this "axiom theme" in higher set theory, this question of whether theories can be "too restrictive." It was Penelope Maddy who, as far as I know, conceived this theme; she also carried it out, as have Peter Koellner and J. D. Hamkins (among others). However, Hamkins rejoined the restriction theme quite subversively, and I think his understanding of the issue currently predominates.
On the other hand, just the other day I came across this puzzling remark:
Berkeley cardinals are so strong, even more than Reinhardt cardinals, that they contradict ZFC.
Kunen's theorem shows when a large large cardinal axiom is too large, so large that it contradicts ZFC.
The template for this talk would seem to be something like ∃κ("ZFC + ∃κ" ⊧ ⊥). Generalized even more, we would say that the inconsistency strength of a theory T is some κ such that "T + ∃κ" ⊧ ⊥. But is that how the above quotes are best read? Is this template enough to show that κ will be (much, much, much) larger than any cardinal whose existence is provable modulo T? Or are there small enough cardinals for any T such that adding them to T would yield an inconsistent model, and so ascent-by-inconsistency-strength is not a reliable method, or at any rate even less reliable than ascent-by-consistency-strength? (Hamkins' rejoinder to Koellner's interpretation-theoretic implementation of the Maddy protocol is that even the consistency-strength hierarchies are not absolutely well-founded or well-ordered.)
The second-order problem. At any rate, on a normalized picture of such things, we would be tempted to equate the inconsistency strength of, say, V = L, with an uncountable measurable cardinal (or zero sharp, or an ω1-Erdős cardinal, or whatever). Or then the inconsistency strength of ZFC is the degenerate supremum of the critical sequence for the Reinhardt embedding. However, I am trying to implement the Maddy protocol in the relevant terms, so we are assuming that any of a certain category of axioms, can be thought of as "too restrictive," viz. by satisfying the inconsistency-strength template at some point or other.
For example,﷼ suppose that there is a universal set V. Now, if every set X has a powerset Y (the axiom), then it is provable that X < Y (the theorem, too often conflated with the axiom). So if V has a powerset, yet not only will that powerset be larger than V, but it will also be an element of V, and in turn, sort of, of itself. So all sorts of shenanigans happen if we assume that there is a universal set and that every set has a powerset. I would surmise that the simplest thing to do would be to just deny that every set has a powerset. I realize there are other reasons for blocking V from being a set (instead of a "proper class"), but none of them are very compelling, ultimately, and at any rate, if we assume that V is powerless, then by a reflection principle we get a powerless cardinal (this also, or even moreso, if V really is taken for a proper class, reflected down into a set form).
I would imagine that powerless cardinals satisfy ∃κ("ZFC + ∃κ" ⊧ ⊥), though whether they do so in fairly direct counterpoint to the initial degenerate Reinhardt, or well beyond even Berkeleys as so far investigated, I don't know. I mean we can perform a "magic trick" and just define κ here not only in terms of inconsistency strength, but also in terms of model-theoretic strength as usual, i.e. every antinomian cardinal otherwise models a restricted(!) version of whichever axiom system it otherwise violates, or is worldly in some way. (Actually, the bare worldliness template is kind of a "mirror image" of the antinomy template, here.) Moreover, I suspect that powerless cardinals would be inaccessible from below, on pain of iterating the powerset operation to their limit allowing us to then represent all the elements of the powerless κ so as to represent all the subsets of κ, and thence form the set of all those subsets, etc.ℙ
The broader point is that we have to differentiate between universal generalizations and universal specifications. A general universal claim ranges over everything generally (or even generically!), whereas a particular universal claim quantifies without restriction over haecceities. V = L, the normal axiom of choice, or the powerset axiom, universally specify all the sets: "All sets are constructible," "All sets are accompanied by choice sets," "Any set is accompanied by a set of all its subsets" (and then so too foundation, singleton, and pairing, perhaps; or even extensionality (just think of how holding extensionality too fast wipes out all the ur-elements)). Is it these universal specifications that give rise to the inconsistency-strength phenomenon, and which are "overly restrictive" in a Maddyesque manner?
The third-order problem. Another phenomenon in upper set theory is the formulation of axiom themes like countable/dependent choice, or Paul Corazza's abridged replacement schemes. These amount to confining all the specified sets produced by the axiom in question, below a relevant limit set, e.g. saying that choice sets are mandated only among countable sets, or that replacement doesn't apply to the embedding function. I imagine, then, that we could just as well make up an axiom theme of X-powersetting, where X is the first powerless set, whatever that is chosen to be, e.g. the countable powerset axiom would say that we can form a set of all subsets of a countable set, but not necessarily of any uncountable set thusly, etc.
Is that the lesson to be learned? That every axiom whose quantifier aspect codes for universal specification, is not only "restrictive" vs. the entire universe of sets, but therefore restricted itself below specified limit sets? Again, then, we would have to speak of sets "so large" that it is impossible to form a singleton over them, or a pairset with them, maybe. Or "so large" that they do not conform to well-foundation. (Maddy thought that we could get by with iterating "isomorphism types" just for well-founded sets, in order to satisfy her maxims; but here we object, claiming that there is "probably" a way to define a very large cardinal such that it is not well-founded, and yet it is larger than anything normally imagined otherwise in such terms; or at least such obscure beings dwell in their own universe in a set-theoretic multiverse, with incommensurable weights.)
Without committing ourselves too much to every claim about which axioms/schemes are problematic universal specifiers, we just ask whether any such axiom is best reformulated as an X-example of its own theme, where X is some specific set in the ascending transet. So in other words, rather than have these axioms meant to range over the entirety of V, we just say that they each hold up to some point, at which point we are impelled to come up with a further axiom system to solve the problems that arise as we ascend(!). So perhaps, say, the determinacy axiom in some form is appropriate as of the point where the normal choice axiom leaves off, yet there must be some much greater object beyond the determinacy realm, one that overpowers that realm, so to speak; and so on and on, onward and upward, forever and ever, amen.
... The fourth-order problem. There seems to be a "quick proof" of the X-likeness thesis just outlined, here. Consider Fitch's paradox of knowability. Now, consider the set of all knowable sets, (for "episteme"). If is itself knowable, then it would be an element of itself. So now every set in the ascending transet is, as such, well-founded. So at least the ascending fragment of is not an element of itself, i.e. is not as such knowable. So then it seems as if we are never going to be in a position to know that we know all the axioms we would need to know, to know as much about all of transcension across V as we would like. But then the inscrutability of transforms (especially in terms of epistemic history, the physical history of the mathematical community) so that we should never expect any such procedural axiom to have unrestricted validity with respect to upwards specification.
Or, at least, that is the conclusion I would be aiming for. Does the inconsistency-strength phenomenon testify against overly "final" assessments of various set theories, in such a way as lends itself to looking at the universe of sets through the lens of ? Or even: is the theory of itself, so to say, such as to imply the inconsistency-strength theory?
﷼Alternatively, consider cofounded sets x = {x} or transfounded sets {... A2 ∈ A1 ∈ A}. It is difficult for me to conceive of them as having proper subsets (or, for example, how are a Quine atom's subsets not all improper?) (note that by "proper" I mean the technical feature of a subset not being a copyset of the base), or even subsets at all (or they might be represented as having only fuzzy subsets?). For example, x being an element of itself, then there is a subset of x with x as its sole element, yet that turns out just to be x again (vs. extensionality, say), and so on. —Elsewhere in the main text, we mention the possibility of large cardinality via cofoundation/transfoundation, but otherwise we do not develop the theme in much detail (but c.f. Golshani, et. al. regarding the relationship between the Kunen wall and the foundation problem).
ℙA quick pair of notes: see this Victoria Gitman, et. al. paper on ZFC minus the powerset axiom for details on what happens to V when that axiom is jettisoned or at least compromised. For the possibility of inflating the Continuum to a proper class, and so via Easton's theorem conflating the powersets of all the alephs, see (starting on pg. 34 but esp. from pg. 206) this broader paper on set theory. Note also that the Kunen wall technically occurs as of a structure most-often denoted by {j: Vλ+2 → Vλ+2}, where λ + 2 = 22λ, i.e. the double powerset of whatever λ is supposed to be. So I do wonder what happens to the Kunen wall in the absence of the powerset axiom being used in its mortar.
In consideration of what is said in the main text about universal specifications, let us interpret {j: V → V} proper as such a universal specifier, i.e. it is the assertion that {j: Vx → Vx} exists for every x, including those from the moat of the Kunen castle onward. But then this becomes the assertion that for every set, there is an embedding-theoretic set above it, formed by an elementary embedding indexed in part by the local base set, and so on. But once again, then, applying the antinomy template, we would "find" that there ought to be some set so large that it is not only beyond the reach of any lower-themed embedding, but is itself such as to not give rise to the relevant kind of embedding. In other words, we can reject {j: V → V} itself as "too restrictive," then, it seems (it rules out the existence of embedding-transcendent sets).
Closing remark. From all of what has just been said, we find that we can identify the first infinite set (whether it be an aleph/omega, or something else) with the inconsistency strength number for finitism. This is arguably "trivial," however: letting finitism = T and the first level of infinity = Ж, then perforce "T + ∃Ж" ⊧ ⊥, namely ⊥ = "There is, and there is not, an infinite set."
