Reflection principles are a good starting point, but applying them, well... Although worldly cardinals are the "smallest type" of ZFC-unprovable cardinals, let's jump up to the uncountable inaccessibles. The classical argument is that *V* is (strongly) inaccessible, i.e. it is not the successor of any number, it is cofinal with itself, and is not the powerset of any smaller set (indeed, *V* itself is not quite a set at all). Therefore there is a limit set that is cofinal with itself and not the powerset of any smaller set. So far so good. But notice that 0 and $\aleph_0$ are equal to their own cofinalities, and are not powersets of smaller sets, and can be construed as limits. So don't they satisfy the reflection principle, here? You might object, "But yeah, we're talking about *uncountable* inaccessibles." So you would have *V* be uncountable, too. However, suppose you adopted ZFC sans replacement. In that event, $\aleph_{\omega}$ counts as a large cardinal (its existence is unprovable in ZFC sans replacement). More importantly, then, it is a shadow of *V* in that compromised instance of ZFC. But since $cf(\aleph_{\omega}) = \aleph_0$), we are in a position where *V* is not quite inaccessible, after all, and in fact is not even perfectly uncountable as such, either. At any rate, the *V*-reflection argument for, "There are inaccessibles," must be amended to, "There are uncountable inaccessibles," but at this point the justification of that assertion has been much weakened.

There's a deeper problem with reflection principles, though, which can be highlighted by formulating them in Ackermann's terms. According to Ackermann, reflection principles are grounded in the ultimate principle, "Absolute infinity is unknowable." However, knowledge is justified true belief (plus some Gettier-dimensional quality). So are we building our theory of set-theoretic justification on a concept of what is *not* justified? That seems wrong, to me.

Now Gödel spoke of intrinsic and extrinsic justification, here. The former roughly translates Kant's doctrine of analytic knowledge. Analytic knowledge is grounded in an unfolding of a concept's content. So Gödel said that if we unfold the iterative concept of set, we will have justified (true!) beliefs about that concept. In the era of set-theoretic multiverses and logical pluralism, on the other hand, we can talk about unfolding all manner of conceptual contents: as long as the concept we are referring to has an elementhood relation encoded into it, we will be analyzing a concept of sets. Anyway, Penelope Maddy refined Gödel by having us iterate isomorphism types: these are what we are MAXIMIZING as we approach absolute infinity. (She claimed, IIRC, that hyperset axioms don't give us new isomorphism types, so that was why nonwell-founded set theory wasn't justified by MAXIMIZE as such. However, it is difficult for me to believe that, "Hyperset *A* is isomorphic to hyperset *B*," is a false kind of sentence, or that the isomorphism, here, would be absolutely identical to some well-founded kind. So I think Maddy's reasoning is deficient, in this case.) Large-cardinal axioms involve these unfolding isomorphism types, ergo...

On the other hand, Maddy's doctrine of intrinsic justification only seems to justify assertions like, "There is *at least one* uncountable inaccessible." But we know that we can contrive set theories in which there are any number of uncountable inaccessibles: finitely many (for any *n*), transfinitely many (for any $\kappa$), or class-many. Maddy's doctrine is of qualitative, not quantitative, iteration: we have already quantitatively MAXIMIZED, after all, as soon as we have admitted any proper class of sets whatsoever. So though Maddy's doctrine looks like a good way to ground assertions that various *types* of large cardinals exist, it is not clear (to me) that she provides us with any solid way to settle questions like, "How *many* cardinals are there, that map from those types?"^{1}

The arguably dominant method of set-theoretic ontology (nowadays) is, however, to advert to generic consistency, and hold that consistent sets of ontological sentences are justified by the illustration of their consistency. This is a kind of "plenitudinous Platonism." (With the advent of paraconsistent set theory, granted, we open the doors to a plenitudinous Platonism that does *not* depend on illustrations of ontological consistency. But that is an aside...) The prevailing case of this doctrine is reference to a set-theoretic multiverse, where the sets of justifiable ontological sentences are converted into the various universes. Inasmuch as Kant originally identified analytic knowledge with a kind of knowledge based on the laws of identity and noncontradiction, we seem to have in the consistency-theoretic method of ontology, the perfect crystallization of the notion of "intrinsic justification" (as analytic).^{2}

All that being said, there is a different way to apply the concept of intrinsic justification, to the issue at hand. And this is to unfold the content, not of the iterative (or any other such) concept of *sets*, but of the concept of *justification* itself. Granted, the furthest along that we've gone here, in the mainstream, is justification logic. But it turns out that we can very easily and strongly justify a number of large-cardinal axioms, in light of justification logic as such.

Case 1. Have there be an axiomatic set theory involving justification logic *simpliciter*; call this "jZFC." For every such theory *T*, there is a relatively worldly cardinal, i.e. at least one $\kappa$ such that $V_{\kappa}$ is a model of *T*. But jZFC is intrinsically justified. Accordingly, the axiom of a worldly cardinal for jZFC inherits this intrinsic justification. (Reasoning: model theory itself justifies various mathematical assertions. So a set theory of model-theoretic justification, inherits the justificatory force of model theory.) Therefore, the axiom, "There is at least one jZFC-worldly cardinal," is as justified as can be.

Case 2. In proof-theoretic ordinal analysis, there are some proof-theoretic ordinals that are to be marked out by using "ordinal collapsing functions." These functions take other large countable ordinals (exceeding the proof-theoretic threshold) for their inputs. In turn, these large countable ordinals can be characterized as mirrors of various genuinely large cardinals. So if jZFC has a proof-theoretic ordinal assigned to it (I see no reason to assume otherwise), then we can reason upwards from that to a jZFC-relevant large countable ordinal, and from *there* to a genuine large jZFC-relevant cardinal. I have next to no idea what other properties such a cardinal would have; but the assertion that such a cardinal exists, seems well-justified by the reasoning at hand.

Case 3. Use an *infinitary* justification logic to background jZFC. There are characterizations of large cardinals that can be designed by reference to matters of normal infinitary logic, e.g. weakly or strongly compact cardinals. So there should be such large-cardinal characterizations available in terms of jZFC. Again, these axioms, as expressions of the concept of justification itself, seem to be intrinsically justified, very strongly, as such. Also, since (for example) strongly compact cardinals "oversee" measurable ones, the jZFC-relevant strongly compact cardinals then ground assertions of measurable cardinals in general, and some bracketing of the, "How many?" question, there, in particular.

I have an idea for justifying axioms of Reinhardt sets, too, in this context, but it depends on disengaging too much with established set theory, to explain as of now. I'll just leave you with the possibility that jZFC could be invoked to justify^{3} asserting the existence of Reinhardt sets, then.

^{1}_{Admittedly, one could probably make up some sort of MAXIMIZE-dimensional reason to move from, "The various types of large cardinals exist," to, "There are class-many of each type." Also, some large $\kappa$ are such that they encode for a lower type of large cardinals, to the effect that we can infer the existence of $\kappa$-many cardinals of the lower type, given below $\kappa$. E.g., since the least measurable $\kappa$ is strongly inaccessible, we get $\kappa$-many strongly inaccessibles below the least measurable.}

^{2}_{On the flip side: if the Kant-Frege thesis that "existence is not a (first-order) predicate" is true, that seems to rule out this talk of justifying an existence claim using the consistency-theoretic method. This is because analytic justification is, therefore, never existential justification as such: all existence claims are synthetic. Personally, though, I wonder whether we might say: it is amiss to speak of existence being analytically true of other things, but is it amiss to speak of other things being analytically true of existence? Or what of an assertion such as, "Existence exists"? I suspect, that is, that we could justify large-cardinal axioms as analytic of the concept of existence: they are forms of existence, that themselves exist. (How are they to be such forms? But remember how Conway describes the meaning of $2^{\aleph_0}$: every real number is individuated by countably many subindexes. In other words, the greater and greater infinite sets, allow us to individuate more and more kinds of otherwise finite numbers. If (as Quine said) to be is to be the value of a bound variable, then the further and further ways of having values and bound variables, are further and further ways "for" things to exist. (Or: if the logic of existence is the logic of existential quantification...) Presto: the higher and higher cardinals are manifestations of higher and higher forms of existence itself, hence can be taken for analytic truths of pure existence. (Going a little sideways, we can use this idea about analytic ontology to "explain" the ontological argument: it is not that existence is analytically true of God, but that God, representative of a form of existence (the created/uncreated dichotomy), is therefore analytically true of existence. (Hence ironically, though Aquinas disavowed the ontological argument, his own claim that God is somehow "subsistent being itself" is tantamount to a recapitulation of that argument, after all.))}

^{3}_{I think we have to differentiate between what jZFC can prove, and what it can justify. By definition, essentially, jZFC cannot for example prove that there is a relatively worldly cardinal. However, it does seem as if it can justify this claim. Now, in justification logic, there are all kinds of second-order questions that come up; modulo jZFC, we have questions like, "Is the assertion, 'X has been proved,' itself justified? And can I prove assertions like, 'X is justified'?" But I don't really know the answers to these questions.}

12more comments