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There are a number of informal heuristic arguments for the consistency of ZFC, enough that I am happy enough to believe that ZFC is consistent. This is true for even some of the more tame large cardinal axioms, like the existence of an infinite number of Grothendieck universes.

Are there any such heuristic arguments for the existence of Vopenka cardinals or huge cardinals? I'd very much like to believe them, mainly because they simplify a great deal of trouble one has to go through when working with accessible categories and localization (every localizer is accessible on a presheaf category, for instance).

For Vopenka's principle, the category-theoretic definition is that every full complete (cocomplete) subcategory of a locally presentable category is reflective (coreflective). This seems rather unintuitive to me (and I don't even understand the model-theoretic definition of Vopenka's principle).

What reason is there to believe that ZFC+VP (or ZFC+HC, which implies the consistency of VP) is consistent? Obviously, I am willing to accept heuristic or informal arguments (since a formal proof is impossible).

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Harry, what are the informal heuristic arguments for the consistency of ZFC? (A reference would be great.) –  danseetea Jun 24 '10 at 22:55
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You'll notice that I never said I've actually read any of them. =D! –  Harry Gindi Jun 25 '10 at 0:44

6 Answers 6

up vote 17 down vote accepted

Most of the arguments previously presented take a set-theoretic/logical point of view and apply to large cardinal axioms in general. There's a lot of good stuff there, but I think there are additional things to be said about Vopěnka's principle specifically from a category-theoretic point of view.

One formulation of Vopěnka's principle (which is the one that I'm used to calling "the" category-theoretic definition, and the one used as the definition in Adamek&Rosicky's book, although there are many category-theoretic statements equivalent to VP) is that there does not exist a large (= proper-class-sized) full discrete (= having no nonidentity morphims between its objects) subcategory of any locally presentable category. I think there is a good argument to be made for the naturalness of this from a category-theoretic perspective.

To explain why, let me back up a bit. To a category theorist of a certain philosophical bent, one thing that category theory teaches us is to avoid talking about equalities between objects of a category, rather than isomorphism. For instance, in doing group theory, we never talk about when two groups are equal, only when they are isomorphic. Likewise in doing topology, we never talk about when two spaces are equal, only when they are homeomorphic. Once you get used to this, it starts to feel like an accident that it even makes sense to ask whether two groups are equal, rather than merely isomorphic. And in fact, it is an accident, or at least dependent on the particular choice of axioms for a set-theoretic foundation; one can give other axiomatizations of set theory, provably equivalent to ZFC, in which it doesn't make sense to ask whether two sets are equal, only whether two elements of a given ambient set are equal. These are sometimes called "categorial" set theories, since the first example was Lawvere's ETCS which axiomatizes the category of sets, but I prefer to call them structural set theories, since there are other versions, like SEAR, which don't require any category theory.

Now there do exist categories in which it does make sense to talk about "equality" of objects. For instance, any set X can be regarded as a discrete category $X_d$, whose objects are the elements of X and in which the only morphisms are identities. Moreover, a category is equivalent to one of the form $X_d$, for some set X, iff it is both a groupoid and a preorder, i.e. every morphism is invertible and any parallel pair of morphisms are equal. I call such a category a "discrete category," although some people use that only for the stricter notion of a category isomorphic to some $X_d$. So it becomes tempting to think that one might instead consider "category" to be a fundamental notion, and define "set" to mean a discrete category.

Unfortunately, however, what I wrote in the previous paragraph is false: a category is equivalent to one of the form $X_d$, for some set X, iff it is a groupoid and a preorder and small. We can just as well construct a category $X_d$ when X is a proper class, and it will of course still be discrete. In fact, just as a set is the same thing as a small discrete category, a proper class is the same thing as a large discrete category. However, this feels kind of bizarre, because the large categories that arise in practice are almost never of the sort that admit a meaningful notion of "equality" between their objects, and in particular they are almost never discrete. Consider the categories of groups, or rings, or topological spaces, or sets for that matter. Outside of set theory, proper classes usually only arise as the class of objects of some large category, which is almost never discrete. The world would make much more sense, from a category-theoretic point of view, if there were no such things as proper classes, a.k.a. discrete large categories --- then we could define "set" to mean "discrete category" and life would be beautiful.

Unfortunately, we can't have large categories without having large discrete categories, at least not without restricting the rest of mathematics fairly severly. This is obviously true if we found mathematics on ZFC or NBG or some other traditional "membership-based" or "material" set theory, since there we need a proper class of objects before we can even define a large category. But it's also true if we use a structural set theory, since there are a few naturally and structurally defined large categories that are discrete, such as the category of well-orderings and all isomorphisms between them (the core of the full subcategory of Poset on the well-orderings).

Thus Vopěnka's principle, as I stated it above, is a weakened version of the thesis that large discrete categories don't exist: it says that at least they can't exist as full subcategories of locally presentable categories. Since locally presentable categories are otherwise very well-behaved, this is at least reasonable to hope for. In fact, from this perspective, if Vopěnka's principle turns out to be inconsistent with ZFC, then maybe it is ZFC that is at fault! (-:

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It should be noted that Petr Vopěnka himself did not believe in the principle! Here is the story, taken from Adámek and Rosický Locally Presentable and Accessible Categories (p. 278-279).

The story of Vopěnka's principle (as related to the authors by Petr Vopěnka) is that of a practical joke which misfired: In the 1960's P. Vopěnka was repelled by the multitude of large cardinals which emerged in set theory. When he constructed, in collaboration with Z. Hedrlín and A. Pultr, a rigid graph on every set (see Lemma 2.64), he came to the conclusion that, with some more effort, a large rigid class of graphs must surely be also constructible. He then decided to tease set-theorists: he introduced a new principle (known today as Vopěnka's principle), and proved some consequences concerning large cardinals. He hoped that some set-theorists would continue this line of research (which they did) until somebody showed that the principle was nonsense. However the latter never materialized — after a number of unsuccessful attempts at constructing a large rigid class of graphs, Vopěnka's principle received its name from Vopěnka's disciples. One of them, T. J. Jech, made Vopěnka's principle widely known.

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William Reinhardt gave heuristic reasons for some very large cardinal axioms in a paper in the proceedings of the 1967 UCLA set theory meeting. I don't know whether he considered cardinals as large as the ones you want, but some of the ideas there might be useful for you. (Disclaimer: I'm not at all convinced that such large cardinals exist. My belief in their consistency is based on the fact that very smart people, like Jack Silver, have looked seriously for inconsistencies and haven't found any.)

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Do you happen to know where to find this paper (or perhaps the title)? –  Harry Gindi Jun 24 '10 at 2:09
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Here's the header from the MathSciNet review: MR0401475 (53 #5302) Reinhardt, W. N. Remarks on reflection principles, large cardinals, and elementary embeddings. Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967), pp. 189--205. Amer. Math. Soc., Providence, R. I., 1974. –  Andreas Blass Jun 24 '10 at 2:17
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Andreas, I worry about the very-smart-people argument. After all, before Wiles, very smart people had looked seriously and been unable to refute FLT, but $\neg$FLT turned out to be inconsistent anyway... –  Joel David Hamkins Jun 24 '10 at 4:00
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Joel, I agree that the smart-people argument is not very strong, and I should have phrased my answer to make that clear. I didn't mean "I believe that very large cardinals are consistent, and the reason is ..."; rather I meant "To the extent that I believe that very large cardinals are consistent, the only reason I have is ...." In the case of small large cardinals (inaccessible, or even indescribable of various levels), I think the reflection idea gives some additional plausibility to the axioms (though I don't think it makes them "clearly true"); I don't see that for measurables and up. –  Andreas Blass Jun 24 '10 at 6:43
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Without implying anything about my personal beliefs, I'm surprised no one has yet mentioned Penelope Maddy's pair of papers called "Believing the Axioms" (I and II), which I enjoyed as a discussion of reasons to believe (or not believe) in various axioms beyond ZFC. –  Mike Shulman Jun 25 '10 at 5:22

Because of Goedel's Incompleteness Theorems, we know that we cannot describe a complete axiomatization of mathematics. Any proposed axiomatization $T$, if consistent, will be unable to prove the principle Con(T) asserting that $T$ itself is consistent, although we have reason to desire this principle once we have committed ourselves to $T$. Adding the consistency principle Con(T) simply puts off the question to Con(T+Con(T)), and so on, in a process that proceeds into the transfinite.

Thus, we come to know that there should be a transfinite tower of theories above our favorite theories, transcending them in consistency strength. The incompleteness theorems imply that there is a tower of theories above PA, above ZFC, each level transcending the consistency strength of the prior levels.

How fortunate and wonderful that we have also independently come upon such a tower of theories: the large cardinal hierarchy. Numerous large cardinal concepts arose very early in set theory, from the time of Cantor, before Goedel's theorems and before the notion of consistency strength was formulated. These large cardinal concepts arose from natural set-theoretic questions in infinite combinatorics: Can there be a regular limit cardinal? Can there be a countably-complete measure measuring all subsets of a set? Does every $\kappa$-complete filter on a set extend to a $\kappa$-complete ultrafilter? And so on.

Eventually, it was realized that these large cardinal notions separate into a very tall hierarchy, with the property that from the larger cardinals, one can prove the consistency of the smaller cardinals. For example, if $\kappa$ is the least Mahlo cardinal, then the universe $H_\kappa$ is a model of ZFC + there is a stationary proper class of inaccessible cardinals + there are no Mahlo cardinals. If $\delta$ is the least measurable cardinal, then $H_\delta$ satisfies ZFC + there are a proper class of Ramsey cardinals, but no measurable cardinal.

Thus, the large cardinal hierarchy provides exactly the tower of theories, whose levels transcend consistency strength, that we knew should exist. And it does so in a way that is mathematically robust and interesting, with its foundations arising, not in some syntactic diagonalization, but in mathematically fulfilling and meaningful questions in infinite combinatorics.

The case of Vopenka's principle is just like this. VP is a large cardinal axiom at the higher end of the large cardinal hierarchy, implying the consistency of the existence of supercompact cardinals, say, which are far stronger than strong cardinals, which imply entire towers of measurable cardinals, which imply numerous Ramsey cardinals and so on down the line.

Illustrating the essential large cardinal nature, the VP axiom is elegantly stated: for every proper class sequence $\langle M_\alpha | \alpha\in\text{ORD}\rangle$ of first order structures, there is a pair of ordinals $\alpha\lt\beta$ for which $M_\alpha$ embeds elementarily into $M_\beta$. (It is equivalently stated in terms just of graphs, if you like.) It's simple and clear---beautiful! And the consequences are far-reaching and often profound, as you have observed in category theory, in the way that VP implies that the set-theoretic universe is regular and organized.

These are the reasons you should be attracted to Vopenka's principle. It is an elegant combinatorial principle, with far-reaching consequences that interest you, which has not yet been refuted.

In contrast, I find the philosophical heuristics that seek to justify the large cardinal axioms, on the grounds of reflection or some other means, to be so much hot air ultimately unsatisfying. These arguments are not mathematically sound, and cannot be made to be, by the Incompleteness Theorems. Philosophically, they seem much more like rationalizations after the fact. For example, even at the much lower (and therefore seemingly easier-to-justify) level of inaccessible cardinals, one sometimes hears an appeal to reflection type views, that since we have no definable unbounded map from a set into the ordinals, that there should be a level $V_\kappa$ of the universe also with this feature, and that such a level would be inaccessible cardinal. Of course, the conclusion outstrips the argument, with the conclusion seeming to justify at most $V_\kappa\models$ZFC, which is a weaker notion, and the meta-reflection principle appealed to amounts anyway to a large cardinal principle of its own.

Ultimately, we must recognize the uncertain nature of all our mathematical enterprise. As our hypotheses rise higher in the large cardinal hierarchy, we must become less sure of consistency---perhaps they will be shown to be inconsistent. This issue arises even at the lowest levels of our mathematical axiomatizations, for we may find at any time (as mentioned in a recent MO question) that even PA is inconsistent. As Woodin says, we all have in our minds the image of a railway line, lined by a sequence of telegraph poles, proceeding into infinity; but when the physicists tell us that the universe is finite, we realize that this picture is pure imagination. Perhaps it is simply inconsistent? So skepticism about consistency has nothing especially to do with the infinite.

Meanwhile, the large cardinal axioms are fascinating and have fascinating consequences. Let's seek out the boundary of consistency, with an attitude tempered by the realization that we may find inconsistency.

In summary, we cannot ever be sure that our axioms are consistent, and we know that above the mathematical theory about which we may be sure, there is a tall tower of theories whose levels transcend consistency. Among them are fascinating theories that are elegantly stated with far-reaching consequences, and which we have not yet refuted. So let's study them! Let's find the boundary between consistency and inconsistency!

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As a historical footnote, I'd like to mention that arguments for incompleteness, in the form "if an inaccessible $\kappa$ exists then its existence is not provable" (because $V_\kappa$ satisfies "there is no inaccessible") actually preceded Gödel's theorems. Kuratowski gave such an argument in 1925 and Zermelo in 1928, though neither was rigorous by today's standards. –  John Stillwell Jun 24 '10 at 3:26
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This is bound to be stupid, but when $M_\alpha$ is a graph, what does $M_\alpha \models \varphi(\vec{x})$ mean? Also, what is $\phi$, and what are its source and target (if that even makes any sense)? –  Harry Gindi Jun 24 '10 at 5:19
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Joel, your remark about graphs instead of arbitrary structures is nice for non-logicians. An alternative way to be nice to them is to replace "elementary embedding" with mere "embedding". (It makes no real difference, since you can Skolemize the structures.) But I don't see that both sorts of niceness can be done simultaneously. Does anyone know an equivalent formulation of Vopenka's principle that uses mere embeddings but for a "simple sort of structure (simpler than a graph plus all its Skolem functions)? –  Andreas Blass Jun 24 '10 at 6:49
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Harry, when $G$ is a graph, then $G\models\varphi$ simply refers to first-order satisfaction for assertions $\varphi$ in the language of graph theory, where you can quantify over vertices and use the edge-relation. You can express things like "G is triangle-free" and "G has total diameter 5" this way, but other assertions, such as "G is connected," are not first order expressible. Andreas, that is a very interesting point. –  Joel David Hamkins Jun 24 '10 at 12:08
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I think I shall delete those words, since I surely don't mean to offend anyone. I just meant to express my frustration with the justifications that are sometimes provided for large cardinals on the basis of reflection ideas, in those instances when to the contrary they are attempts to justify our confidence in the axioms. It is, of course, the reflection principles themselves, suitably formalized, that carry a large cardinal interest. –  Joel David Hamkins Jun 25 '10 at 1:31

Here is a practical argument. Set theorists like to solve various problems and large cardinals help as either (rarely) they imply a positive answer or (more usually) their consistency implies the consistency of a positive answer. Under "positive" I mean an answer that is not a counterexample, an answer that does not give an object with strange properties. For example, determinacy at the n-th level of the projective hierarchy solves most problems on sets of that level positively (if blah are sets of that level then blah blah vs there is a sequence of blah sets for which this and this hold but this and this do not) and said determinacy is equiconsistent to the existence of n Woodin cardinals.

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The standard heuristic argument for large cardinal axioms (including huge cardinals) is the reflection principle. The intuitive idea is that $V$ is "absolutely infinite" and so cannot be defined as the collection which satisfies $\varphi$; there will always be some smaller $V_\alpha$ that already satisfies $\varphi$. See the paper Higher Order Reflection Principles by M. Victoria Marshall R. for more details.

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Does this heuristic fail for Reinhardt cardinals, which are known to be inconsistent with ZFC? –  Harry Gindi Jun 24 '10 at 1:48
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I think you're asking if every plausible-looking reflection principle can be trusted. The answer is no; there is a reflection principle corresponding to Reinhardt cardinals (this may even have motivated the definition of Reinhardt cardinals in the first place, but I'm unsure of my history), and in fact there are other reflection principles which are even more easily shown to be inconsistent. So why trust a heuristic that doesn't always work? There have been attempts to delimit the acceptable reflection principles; see people.fas.harvard.edu/~koellner/papers/ORP_final.pdf –  Timothy Chow Jun 24 '10 at 2:18
    
The link people.fas.harvard.edu/~koellner/papers/ORP_final.pdf is no longer working (at least I get "access forbidden 403" there). Is there another place to download this pdf from? I'd be curious about the contents. Thanks –  Desiato Oct 10 at 1:09
    

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