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In different books one can find different implicit definitions for a large cardinal axiom.

My question is that which one of these definitions are more popular or standard amongst set theorists?

Any reference for an explicit definition of a large cardinal axiom is welcome.

  • I suggest asking this on math.stackexchange.com if this question is closed. – David Roberts Jan 24 '14 at 05:08
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    I think this is actually an important, if soft, question worthy of discussion by set-theorists with some interest in the subject. There is at least one formal definition that I know of offered by Woodin in part II of his article on CH in the AMS. I don't know if he himself considers the definition there satisfactory, but there doesn't seem to be any consensus on the topic (as far as I can tell). – Everett Piper Jan 24 '14 at 05:16
  • @EverettPiper: If there is no consensus on the topic, then the question has no answer and probably doesn't belong here. – Steven Landsburg Jan 24 '14 at 05:41
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    @StevenLandsburg: Couldn't it move toward a consensus? Sites like this provide a deeper and at the same time broader understanding of topics informed by experts working in the field, even if that understanding borders on what might be considered more philosophical concerns. If the only "definitions" of large cardinal are implicitly understood, so be it. But certainly we could benefit from having a single source collecting together some of these implicit "definitions", right? – Everett Piper Jan 24 '14 at 05:50
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    I would be interested to learn what are the various technical definitions that have been proposed, such as the one mentioned by Everett, and I consider this question on-topic for MO. – Joel David Hamkins Jan 24 '14 at 13:37
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    And just to clarify, of course we are not looking for examples of large cardinal axioms, but rather a formal definition of what it means for an axiom to be a large cardinal axiom. This is a fairly slippery concept, and naive attempts to formalize this are typically inadequate. – Joel David Hamkins Jan 24 '14 at 14:32
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    Sort of like http://mathoverflow.net/questions/104016/what-is-a-choice-principle-really – Asaf Karagila Jan 24 '14 at 14:46
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    I confess that I'll be disappointed (!) if this question has a good answer. I believe that there are certain mathematical concepts that derive their utility from not being formal, and "large cardinal axiom" is, IMO, one of them. Another example is the term "sieve" in number theory. It's so ingrained in most of us that formalizing a definition constitutes "progress" that I think it's good to have a few clear counterexamples lying around. – Timothy Chow Jan 24 '14 at 15:40
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    @EverettPiper: Thanks. Would you please add a link for the paper? –  Jan 24 '14 at 17:56
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    Timothy, I agree with that, but I think the actual project here is to poke holes in the all the proposed answers. – Joel David Hamkins Jan 24 '14 at 18:31

4 Answers4

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Let me give the definition of large cardinals given by Woodin, mentioned by Everett.

Definition. $\exists x\phi(x)$ is a large cardinal axiom, if $\phi(x)$ is a $\Sigma_2$-formula, and as a theorem of ZFC, if $\kappa$ is a cardinal such that $V\models \phi(\kappa),$ then $\kappa$ is strongly inaccessible, and for all forcing notions $P$ of size $<\kappa, V^P\models \phi(\kappa).$

I think an interesting question is: which known large cardinals do not have a $\Sigma_2$-definition?

Asaf Karagila
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Mohammad Golshani
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    Interesting. I always thought of $0^#$ as a large cardinal axiom, or at least Jónsson cardinals. – Asaf Karagila Jan 24 '14 at 15:00
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    I think $0^\sharp$ is considered as a large cardinal axiom, since it implies uncountable cardinals are large cardinals (in the sense of Woodin!!!) in $L$. – Mohammad Golshani Jan 24 '14 at 15:01
  • Yeah, but it certainly doesn't satisfy the given definition. – Asaf Karagila Jan 24 '14 at 15:02
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    Woodin's $\Omega-$conjecture implies large cardinals are linearly well-ordered. – Mohammad Golshani Jan 24 '14 at 15:07
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    Also for the interesting question part, "There exists a class of ..." is generally not $\Sigma_2$. Even though it is often preserved by any set forcing. Rowbottom and Jónsson cardinals don't have to be strongly inaccessible, so they don't satisfy those definitions either, by the way. – Asaf Karagila Jan 24 '14 at 15:38
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    Does Woodin explain why preserving under small forcings and having $\Sigma_2$ definition are such an important matter in defining the notion of a large cardinal axiom? –  Jan 24 '14 at 17:26
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    This definition does not seem to cover cardinals like supercompact, strong, strongly compact, etc., since these are not $\Sigma_2$ definable. Also, it doesn't cover the worldly cardinals, since they are not inaccessible. – Joel David Hamkins Jan 24 '14 at 18:30
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    To add to Joel's comment: Woodin's definition is in the context of a proper class of Woodin cardinals, so it is definitely not meant to be comprehensive (or to be taken seriously as an attempt to formalize the notion). Rather, it is an attempt to formalize our intuition that the large cardinal hierarchy is indeed well-ordered, by showing that a reasonable subclass is well-ordered under appropriate assumptions. The restriction to $\Sigma_2$ is to make the notions local. Supercompactness, or even strongness are not of this kind; instead, inaccessible cardinals such that (Cont.) – Andrés E. Caicedo Jan 24 '14 at 20:07
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    $V_\kappa$ is a model of "there is a proper class of supercompact cardinals", for instance, form a $\Sigma_2$ class, so the restriction is essentially innocuous. On the other hand, some set theorists are bothered (this is purely a aesthetic matter) by the idea of global large cardinals; so, for example, strong cardinals are perfectly fine, and expected, in inner models, but not as part of the true universe (whatever that is). Inaccessibles with $V_\kappa$ having a proper class of strong cardinals are fine under this view.) The other restriction is just to formalize the idea that we (Cont.) – Andrés E. Caicedo Jan 24 '14 at 20:12
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    have grown accustomed to generalizations of Levy-Solovay. (Of course, this makes the examples we have of large cardinals that do not have a Levy-Solovay theorem interesting for other reasons, as they appear more difficult to handle.) – Andrés E. Caicedo Jan 24 '14 at 20:15
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    Andres, can you point to any publication, either ancient or contemporary, where a set theorist has expressed some disapproval of "global" large cardinals? This seems a little bizarre to me. – Monroe Eskew Jan 25 '14 at 07:44
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    @MonroeEskew I doubt there is anything in print, the comment comes from private conversations. That said, if there is anything in writing, I would begin looking for it on places commenting on Laver's indestructibility theorem. – Andrés E. Caicedo Jan 25 '14 at 16:50
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    Sy Friedman's Inner Model Hypothesis (see http://www.sciencedirect.com/science/article/pii/S0168007212000280) implies the nonexistence of even inaccessibles in the "real" universe, although measurables of arbitrary Mitchell order will exist in inner models; so, to the extent that (as in the cited paper) he makes a philosophical argument for the IMH, does this count? – Noah Schweber Feb 01 '14 at 07:04
  • @NoahS, I don't think that counts, because it goes too far. The IMH contradicts all large cardinals, whereas Andres suggested a position in which $\Sigma_2$ large cardinals are accepted but not more complex ones. – Monroe Eskew Jun 03 '14 at 17:38
  • This is an interesting definition. Could you show me a reference? – Master Jul 24 '19 at 15:30
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While I think I agree with Tim Chow and Joel Hamkins in some of their comments above regarding a single formal definition of what it is to be a large cardinal, I want to suggest that a large cardinal be considered such if it satisfies the following open-ended, semi-formal "definition" (which I draw from history, not a priori ideas about largeness or related notions about the universe of sets (whatever that could mean)). I.e., a set/proposition is a large cardinal notion if it satisfies an open-ended disjunction which includes among its disjuncts the three categories (not intended as the logical notion, just plain ol' English) I mention below.

Historically, large cardinal assumptions seem to fall into one of several categories:

1) Inaccessibility "from below" of some sort.

2) As critical points of elementary embeddings between certain set-theoretic structures.

3) Propositions formalizable in, say, ZFC or some second-order strengthening/alternative, which logically entail the existence of other large cardinal notions.

Per 1: This criterion will obviously capture the weakly and strongly inaccessible large cardinals as closure points under the usual set-theoretic operations. I take it that this informal notion will also give us Mahlo cardinals. I would include in this category indescribable cardinals of various degrees, hence also weakly compact cardinals (though these will fall under both categories 2 and 3 as well).

This inaccessibility requirement also encompasses proof-theoretic considerations. So, for example, statements which entail the existence of models of say ZFC, would fall under this category as well. This would include the iterated consistency hierarchy, the existence of a transitive (set?) model of ZFC, and worldly cardinals and their generalizations.

I think this category might also include $V=L$ as a large cardinal axiom, though there is some conflict with the other 2 categories. I hope to explain this away by proclaiming that a large cardinal assumption satisfies at least one of the three categories above while at the same time not satisfying a plausible negation of any of the three categories (or any further, since the list is intended to be open-ended). I'm not sure this is fair to do, since I'm implicitly importing a consistency constraint, but at this point, I believe it's at least reasonable.

per 2: This particular category encompasses, as far as I can tell, most of the large large cardinal axioms. Since this is pretty standard, I take it that there is no objection to this category as a criterion for "large cardinal". But please do offer any objections in the comments section.

per 3: This last category is intended to encompass assumptions relating to the existence of various kinds of sharps, indiscernibles for various set-theoretic structures, and axioms like $AD^{L(\mathbb{R})}$ and generalizations of these structures.

I think this category will also encompass the existence of certain types of indestructible large cardinals, since many such cardinals have been to shown to exist by assuming the existence of a particular large cardinal and then providing a certain forcing construction.

Again, this criterion may be a little unfair since the proposition $0=1$ implies every set-theoretic statement, including all so-called large cardinals. Thus, I intend to implicitly require a consistency constraint...

Some remarks about the above proposal:

1) In an ideal world, large cardinals would logically imply some regular/predictable structure on the universe below. In particular, I think a particularly desirable feature would be that the assumption in question imposes a linear order on the large cardinals strictly smaller. From an aesthetic point-of-view, this would require that the "identity crisis" phenomenon observed in say, strongly compact cardinals, be eradicated by the assumption in question.

To clarify, I would not reject, at this point, a specific large cardinal assumption that did not "tame" the large cardinals below it as a large cardinal. Rather, I'm more inclined to reject that a specific notion proposed as a large cardinal notion, no longer be considered a large cardinal in the context of assuming another large cardinal if the first is not "tamed" in the context of the second, but virtually everything else is. This is a vague idea, I know. I'm simply trying to isolate which features become forcing invariant under a specific large cardinal assumption.

2) Somewhat contrary to the first remark, I'm inclined to believe that preservation under small forcing (this is Levy-Solovay) is not a feature necessarily shared by all large cardinal assumptions. Although this feature is an empirical phenomenon known for most large cardinals, I'm not yet convinced that it is a characterizing feature of large cardinal assumptions. If the consensus of the community is that Levy-Solovay is somehow an intrinsic or desirable feature for a large cardinal assumption to satisfy, so be it. But I think there are many examples showing that large cardinal assumptions can in fact be susceptible to both large and small forcing notions.

3) Pen Maddy has a very thorough set of principles for proposing new axioms, expressed across two articles, that are at the very least worth reading. I'll add a link when I find it. I encountered these articles a long time ago and even found them compelling from a certain point-of-view. However, I didn't consult them for this answer and I also believe there has been research done since their publication that undermines some (or maybe even most) of those principles.

I invite any and all critiques (as well as additions) to this proposal.

Everett Piper
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    I strongly disagree with any notion in which $V=L$ is a large cardinal axiom. More generally, if $\varphi$ is a statement whose consistency does not exceed that of $\sf ZFC$ then $\varphi$ cannot be a large cardinal axiom. If that would be the case, then since large cardinals make sense in $\sf ZF$ as well, we can say that $\sf AC$ is a large cardinal axiom too, or $\sf DC$ - which is equivalent to the existence of elementary submodels of size $\aleph_0$. These seem to betray the very fundamental idea that a large cardinal is something adding to the strength of the system. – Asaf Karagila Feb 02 '14 at 03:03
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    Asaf, I disagree with your assertion that large cardinal axioms must transcend ZFC. I would find it reasonable to regard the axiom "$\omega_1$ exists" as an incipient large cardinal axiom, even though it is provable in ZFC (but not in ZFC-). The axiom "$\aleph_\omega$ exists" is a large cardinal axiom that is provable in ZFC, but not provable in ZC. In this sense, we could regard both the power set axiom and the replacement (or collection) axioms as proto-large cardinal axioms. But I probably agree with you on $V=L$, since I would think $V\neq L$ is more large-cardinal-like. – Joel David Hamkins Feb 02 '14 at 03:09
  • Dear @Joel, I admit that I haven't really given much thought to large cardinals in $\sf ZFC^-$ or $\sf Z$, or other weak systems. So perhaps there's something to what you're saying. I'll think about it for a few months. – Asaf Karagila Feb 02 '14 at 03:36
  • @Joel and Asaf: given your comments, how would you alter any of the proposed categories? I'm certainly willing to give up the full strength of ZFC, but I'm really not familiar enough with many of its fragments and their respective strength. Is ZFC or its strengthenings simply too strong a base to use when attempting a definition of "large cardinal"? – Everett Piper Feb 02 '14 at 03:46
  • @Joel, in similarity to the "What is a Choice Principle, Really" thread, would you say that $$\lnot\sf AC\rightarrow\exists\kappa\text{ inaccessible}$$ is a large cardinal axiom, or is it some hybrid choice principle-large cardinal axiom? – Asaf Karagila Feb 02 '14 at 04:29
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    Thanks for your answer. Here are links to Penelope Maddy's papers, Believing the Axioms I & Believing the Axioms II. Also her books Defending the Axioms, Naturalism in Mathematics, Realism in Mathematics and Second Philosophy are all interesting and related to the subject. –  Feb 02 '14 at 08:49
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    @JoelDavidHamkins One can consider the axiom of infinity as a primitive large cardinal axiom in the Everett's first category. $\omega$ is somehow inaccessible from finite numbers and has the properties of many ordinary large cardinals. In fact without uncountability assumption $\omega$ is a large cardinal in many senses, e.g. $\omega$ is strongly compact. –  Feb 02 '14 at 09:52
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    @Konrad: In several places large cardinal axioms are called "Strong axioms of infinity" (or variation thereof) because they postulate the existence of uncountable cardinals which have properties similar to $\omega$ (or generalizations of them, in some cases). And I completely agree that the axiom of infinity is a large cardinal axiom when considering $\sf ZFC-Inf$. It increases the consistency strength, and so on and so forth. – Asaf Karagila Feb 02 '14 at 15:40
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A general axiomatic framework for large cardinal axioms has been attempted by Apter, Diprisco, Henle, Swicker in the paper "Filter spaces: towards a unified theory of large cardinal and embedding axioms" (1989, MR0982997) and the continuation "Filter spaces. II. Limit ultraproducts and iterated embeddings" (1989, MR1071798).

François G. Dorais
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I have studied large cardinals for a while. A definition that I read somewhere (although I don't remember exactly where):

A large cardinal axiom is an axiom assuming the existence of a $\kappa$ so that $\kappa \geq \min\{\lambda: \lambda = \aleph_\lambda\}$

As such, I would consider the existence of alef-fixed point to be the "smallest" large cardinal axiom.

Binary198
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    The question is not about how to define large cardinal axioms, but about which ones are more popular of the many existant definitions. – Alex M. Feb 16 '22 at 12:34
  • Oh, sorry. I will then just replace this with the other definition I occasionally use. – Binary198 Feb 16 '22 at 13:37
  • Maybe I'm missing something (possible, since at best I'm only vaguely familiar with some types of large cardinals), but my understanding is that $\lambda \rightarrow \aleph_{\lambda}$ is a normal function (when $\lambda$ is restricted so we're not dealing with non-set classes), and thus has fixed points, and now we can just take the successor cardinal (or exponentiate) one of them. In fact, I've always thought that from a large cardinal perspective this is about the same as using the successor function -- see my comments to this question. – Dave L Renfro Feb 16 '22 at 19:33
  • Yes. I would just say anything above the least alef fixed point is a large cardinal. – Binary198 Feb 16 '22 at 20:42