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Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$).

Their non-existence is consistent with axioms of usual mathematics.

It is provable that some of them don't exist at all.

They show many unusual strange properties.

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These are a part of arguments which could be used against large cardinal axioms, but many set theorists not only believe in the existence of large cardinals, but also refute every statement like $V=L$ which is contradictory to their existence.

What makes large cardinal axioms reasonable enough to add them to set of axioms of usual mathematics? Is there any particular mathematical or philosophical reason which forces/convinces us to accept large cardinal axioms? Is there any fundamental axiom which is philosophically reasonable and implies the necessity of adding large cardinals to mathematics? Is it Reflection Principle which informally says "all properties of the universe $V$ should be reflected in a level of von Neumann's hierarchy" and so because within $\text{ZF-Inf}$ the universe $V$ is infinite we should add large cardinal $\omega$ (which is inaccessible from finite numbers) by accepting large cardinal axiom $\text{Inf}$ and because $V$ is a model of (second order) $\text{ZFC}$ we should accept existence of inaccessible cardinals to reflect this property to $V_{\kappa}$ for $\kappa$ inaccessible and so on?

Question. I am searching for useful mathematical, philosophical,... references which investigate around possible answers of above questions.

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    $\begingroup$ Isn't it a little arrogant to propose some anti-large cardinal axiom like V=L? It's as if you know all there is to know about the structure of the world. Large cardinals are a way of acknowledging our limitations in comparison to the huge complicated world out there. $\endgroup$ Apr 25, 2014 at 3:43
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    $\begingroup$ @Monroe Some have argued that point on the other side. For example, Stephen Simpson compares large cardinal skepticism with religious skepticism. $\endgroup$ Apr 25, 2014 at 4:05
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    $\begingroup$ Nobody forces you to accept anything. If you don't want to accept large cardinal axioms, you don't have to. $\endgroup$
    – Asaf Karagila
    Apr 25, 2014 at 4:15
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    $\begingroup$ While I think large cardinals are fun, I don't feel particularly forced to have an opinion about their existence. It would be way cool if they could be explained in terms of computation. For instance, inaccessible cardinals correspond to type-theoretic universes, and Mahlo cardinals can be seen as a very strong induction principle in type theory. But what about even larger cardinals? Do they have a computational meaning? That would "force" me personally to regard them seriously. $\endgroup$ Apr 25, 2014 at 7:39
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    $\begingroup$ @MonroeEskew That is a common position, and you certainly have excellent company in it. But how do you reconcile such confidence about that with your remarks about human understanding and arrogance? $\endgroup$ Apr 26, 2014 at 13:08

7 Answers 7

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The line of reasoning you mention at the end of your post, firmly in support of large cardinals, was first argued forcefully in

  • W. N. Reinhardt, “Remarks on reflection principles, large cardinals, and elementary embeddings,” Proceedings of Symposia in Pure Mathematics, Vol 13, Part II, 1974, pp. 189-205

and the ideas are further discussed, explained and basically supported in

These articles have now a rather large literature of discussion and criticism in the philosophy of set theory. To get started, you might find further resources on the reading list of my recent course NYU Philosophy of Set Theory. One can now find numerous articles arguing on any given side of each issue.

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  • $\begingroup$ Your answer is quite helpful. Thank you very much. $\endgroup$
    – user47697
    Apr 25, 2014 at 11:19
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There are (possibly) two questions here:

  1. Why should we believe that large cardinal axioms are consistent?

  2. Why, if we believe that large cardinal axioms are consistent, should we believe that they are true?

Here are some reasons for believing that large cardinal axioms are consistent (or at least, that small large cardinal axioms that have been studied for a long time are consistent.)

First, there is the empirical fact no one has published a proof of a contradiction from the assumption $\mathsf{ZFC} + {}$"there is an inaccessible cardinal" (for example) despite a long period of study in which many theorems have been proved from this assumption. Although some large cardinal hypotheses (such as Reinhardt cardinals) have turned out to be inconsistent, this was discovered relatively quickly, in the period during which most people were still skeptical of them.

Second, there is "fine structure" which gives canonical models for the smaller large cardinal axioms (so far, up to Woodin cardinals and a bit further.) It seems reasonable to expect that a systematic study of the structure of the models of a theory would eventually reveal the inconsistency of the theory if it were inconsistent, and this has not happened yet.

For question 2, let us now assume (informally, for the sake of non-mathematical argument) that large cardinal axioms are consistent. Why should we then believe that they are true? Most people find it natural to believe the assumptions that they use in their day-to-day work, so this question is closely related to the question of which axioms people should use. Of course, the answer will depend on what types of theorems they want to prove. In most areas of mathematical research, $\mathsf{ZFC}$ seems to be sufficient in a practical sense and there does not seem (to me) to be a compelling argument that people working in these areas should use, or believe, any kind of axiom beyond $\mathsf{ZFC}$.

So perhaps the question should be revised to "why should mathematicians who want to prove theorems beyond $\mathsf{ZFC}$ use large cardinal axioms, instead of alternatives such as $V=L$?" A practical answer is that doing so allows us to prove lots of interesting theorems. Suppose that I assume $\mathsf{ZFC} + {}$"there is a measurable cardinal" and you assume "$\mathsf{ZFC} + V=L$." Then for every theorem that you prove, I could have proved (if I were clever enough) a corresponding theorem of the form $L \models \ldots.$ On the other hand, I may have the opportunity to discover an interesting theorem about measurable cardinals that you do not have the opportunity to discover (unless you investigate countable transitive models with measurable cardinals, which seems like an unnatural thing to do if you believe that $V=L$, even though it is presumably formally consistent for you to assume the existence of such models.)

This last point is summarized by the slogan "maximize interpretive power." Many of the points I made above are better made in the following paper. I think that what I wrote here leans toward Steel's viewpoint, but I do not claim to have rendered it faithfully.

Feferman, Solomon; Friedman, Harvey M.; Maddy, Penelope & Steel, John R. (2000). Does mathematics need new axioms? Bulletin of Symbolic Logic 6 (4):401-446.

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    $\begingroup$ Regarding your comment: "Most people find it natural to believe the assumptions that they use in their day-to-day work…" I would venture to guess, from the point of view of human psychology, that one common trait of successful mathematicians is the ability to manage their level of belief in various unproved statements---strengthening it as they attempt to achieve proof; gutting it as they attempt to achieve disproof. $\endgroup$
    – Lee Mosher
    Apr 26, 2014 at 14:21
  • $\begingroup$ @LeeMosher, it’s not just mathematicians. Someone may well assume that a precedent (legal or social or scientific) will be followed, even if they believe the precedent is bad. $\endgroup$
    – Matt F.
    Nov 11, 2021 at 19:29
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This is a personal opinion rather than an answer (in fact another personal opinion of mine is that this kind of question cannot have a meaningful objective answer).

Compare this situation with Euclidean geometry. It is not quite correct to ask whether one should believe in the fifth postulate or not. This is because with the current state of knowledge there is no problem at all to deal with all possible versions of it. And in fact, already situations when the status of the fifth postulate varies from point to point are very well understood.

In set theory also, already state of knowledge is ripe to study if not all, then at least significant amount of possibilities which can arise from various combinations of large cardinal (and several other important) axioms. And in fact it is perfectly meaningful to consider and study mathematical structures which allow for variability of the status of these axioms similarly to the variation of curvature on a geometric surface.

I believe that in such circumstances the question of belief becomes obsolete. It is true that in physics one may believe that the universe is positively or negatively curved, or flat. But this is because we are placed inside this universe. In case of mathematics, we are not placed inside any particular model of set theory, hence we are not forced to choose. Certainly some models are distinguished among the rest by some special properties, like flat geometry is distinguished among the rest of geometries, but that's all one can say I think.

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  • $\begingroup$ I believe that your belief that the question of belief is obsolete is false. Here's why. It's possible that ZFC is consistent but has no ω-model (e.g. proves itself to be Σ[k]-unsound for some fixed k, or proves itself to be arithmetically unsound). In that case, sufficiently large cardinals simply don't exist. So you at least have to believe that ZFC has an ω-model. But why do you believe that? Are you sure that unbounded specification plus replacement do not cause it to prove some false arithmetical sentence, and why? $\endgroup$
    – user21820
    Jan 23 at 17:59
  • $\begingroup$ @user21820 I should certainly separate independence and consistency from each other. When some new principle is proved to be independent from current theory, one can say that it does not make much sense to believe in such principle or its negation. Because usually it is the case that if you believe there is a model validating the new principle then from such model one can construct another one validating negation of that principle. The cases when it is not known whether adding a new axiom leads to contradiction or not are entirely different. About such cases I suppose I cannot say anything. $\endgroup$ Jan 24 at 21:26
  • $\begingroup$ Sorry I don't understand how your comment addresses mine. I agree that we should separate consistency and existence, but are you trying to say that your statement "the question of belief becomes obsolete" is only about consistency? Also, I can't understand how large cardinal axioms would be meaningful unless you believe ZFC has an ω-model, and that's why I asked you why you believe it. That is, there does not seem to be much meaning in asking the question "Is ZFC+A consistent?" as a foundational concern if we already believe ¬A. $\endgroup$
    – user21820
    Jan 24 at 21:37
  • $\begingroup$ @user21820 On the contrary, I said that it is only about independence. If (under some assumptions) we can prove that both ZFC+A and ZFC+¬A have a model, then, I would say, we can simply try to study both models of ZFC+A and models of ZFC+¬A rather than believing that one of those two is "better" in one sense or other. Whether If we don't know about any reasonable assumptions which would imply that ZFC+A is consistent then maybe it makes sense to still believe it on some grounds, I don't know. $\endgroup$ Jan 24 at 22:00
  • $\begingroup$ I think you completely missed my point. If you believe ZFC is consistent, you must believe that ZFC+¬Con(ZFC) has a model. But you have absolutely no reason to study models of ZFC+¬Con(ZFC), because it is meaningless precisely based on your belief. $\endgroup$
    – user21820
    Jan 25 at 14:30
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It is useful in the category theory, in particular, in its applications to algebraic geometry. "Small" categories (whose objects form a set) are much nicer than "large categories" (whose objects mere form a "class", whatever it means). In algebraic geometry one wants to consider categories like Sets, Schemes and so on as small --- and technically it is done using Grothendieck's "Axiom of universe", which is equivalent to existence of strongly inacessible cardinals large than a given cardinal, see http://en.wikipedia.org/wiki/Grothendieck_universe

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    $\begingroup$ I think this is a good reason. Non set-theoretic reasons for believing in set-theoretic concepts are the most compelling, by far. $\endgroup$ May 23, 2014 at 13:39
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This is only a partial answer, but Harvey Friedman has a research program to find concrete $\Pi^0_1$ sentences that are purely combinatorial (i.e., make no reference to concepts from logic such as axioms or formal systems) that can be deduced from a large cardinal axiom and that imply the consistency of a (slightly weaker) large cardinal axiom. The $\Pi^0_1$ statement can of course be partially verified by direct computation, so if you convince yourself that it is true, then the large cardinal axiom helps "explain" why it is true. I believe that Friedman has carried out his program up to and including subtle cardinals; see this post on the Foundations of Mathematics mailing list, for example.

I believe Friedman is optimistic that his program can in principle be carried out for any large cardinal axiom, but at present I believe he has no natural, explicit $\Pi^0_1$ statements that require (say) measurable cardinals to prove.

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  • $\begingroup$ In Friedman's examples, do you know if the consistency of a given large cardinal axiom enough to prove the given $\Pi^0_1$ statements, or does it actually require the existence of a given large cardinal? $\endgroup$ Dec 11, 2014 at 12:28
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    $\begingroup$ @JesseElliott : Usually what is needed is 1-consistency. If you think about it, there's no way that a large cardinal could be required by an arithmetical statement S in the strongest possible sense that its existence is actually implied by S, because there are models of true arithmetic in which there is no inaccessible cardinal. $\endgroup$ Dec 11, 2014 at 18:37
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Let me give an argument, not that we should believe large cardinal axioms or their consistency, but rather that regardless of our belief in consistency we should still be interested in results around them.

First, large cardinals are uniquely useful in analyzing principles of strong consistency strength. That is, we know from experience that there are many principles, with consistency strength greater than $ZFC$, for which large cardinals function as a useful organizational principle. This is especially important if I'm agnostic about the consistency of large cardinals (which I am), since then I'm also agnostic about a bunch of other fairly natural principles and want a nice "yardstick" to organize my knowledge of them.

Even if I actively believe, say, that "there is an inaccessible" is inconsistent with $ZFC$, playing with large cardinals is still useful to me: if I believe inaccessibles are inconsistent with $ZFC$, then I also must believe that "DC + every set of reals is Lebesgue measurable" is inconsistent with $ZF$; the point is, there are reasonably natural philosophical viewpoints which reject inaccessibles - say, believing that the Inner Model Hypothesis is true - which yield, via arguments around large cardinals, philosophical positions against other principles for which no such natural viewpoint is known to exist.

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  • $\begingroup$ Do we have examples of axioms A and B, not obviously about large cardinals, for which Con(A) $\rightarrow$ Con(B) can only / best / most easily be proved via theorems about large cardinals? Such an example would strengthen this argument substantially. $\endgroup$
    – Matt F.
    Apr 26, 2014 at 21:08
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    $\begingroup$ @MattF. An example is that $\mathbf{\Pi}^1_1$-determinacy implies $<\omega^2$-$\mathbf{\Pi}^1_1$-determinacy. The only known proof goes by showing that the first assumption implies the existence of sharps for reals, and that the sharps imply the stronger determinacy assumption. Other examples of such transfer theorems in descriptive set theory are also known at higher consistency strength. See here for more on this. $\endgroup$ Apr 26, 2014 at 21:19
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    $\begingroup$ @MattF. One inaccessible. In $L$. This is a consequence of famous results of Solovay and Shelah. $\endgroup$ Apr 26, 2014 at 22:29
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    $\begingroup$ @MattF. I would ask that you do not delete the comments. I think they may end up being more useful (since they clarify Noah's point) than if a summary is instead posted in Noah's answer (as the subtle difference between existence in $V$ and existence in inner models may be otherwise glossed over on a casual reading). $\endgroup$ Apr 26, 2014 at 22:37
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    $\begingroup$ @MattF., this is the result mentioned in the final paragraph of my answer - the relevant source is "Can you take Solovay's inaccessible away?" (link.springer.com/article/10.1007%2FBF02760522) by Saharon Shelah (specifically, this is the paper that showed that Con(ZF+DC+everything measurable)$\implies$Con(ZFC+inaccessible); the converse direction had been proved by Solovay math.wisc.edu/~miller/old/m873-03/solovay.pdf). Zero sharp in my comment is kind of a red herring; the point is that inaccessibles exactly capture "everything's measurable," and zero sharp is stronger. $\endgroup$ Apr 26, 2014 at 23:48
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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 justify3 asserting the existence of Reinhardt sets, then.

1Admittedly, 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.

2On 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.))

3I 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.

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