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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.

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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 at 5:16
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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 at 5:50
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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 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 at 14:32
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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 at 18:31

4 Answers 4

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?

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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 at 15:01
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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 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? –  user45939 Jan 24 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 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.) –  Andres Caicedo Jan 24 at 20:07

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.

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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 2 at 3:03
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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 2 at 3:09
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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. –  user45939 Feb 2 at 8:49
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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. –  user45939 Feb 2 at 9:52
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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 2 at 15:40

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).

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I don't know if this definition of a large cardinal axiom is too board or not but at least includes almost all axioms which are known as a large cardinal axiom and almost all other statements of set theory with a large cardinal strength.

Definition. A large cardinal axiom is a first order statement $\sigma$ in the language of set theory such that,

$ZFC\vdash Con(ZFC+\sigma)\longrightarrow Con(ZFC+Con(ZFC))$

In the other words a large cardinal axiom is a statement of set theory which is stronger than consistency of ZFC (existence of a model for ZFC) in consistency strength order.

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Doesn't this (ironically) go down a bit too far in the hierarchy? e.g. all the way down to inaccessibles, which is pretty much as far down the large cardinal hierarchy as one can go... –  Steven Stadnicki Feb 1 at 6:10
    
@StevenStadnicki Way below inaccessibility. –  Andres Caicedo Feb 1 at 6:33
    
In particular, any sentence implying $Con(ZFC)$ (over $ZFC$) has this property. This really doesn't seem right (especially since "$Con(ZFC)$" itself is just a $\Pi^0_1$ statement, and so shouldn't be a large cardinal axiom). –  Noah S Feb 1 at 7:05
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This proposal has the property that if $\sigma$ is a large cardinal axiom, then so is $\sigma\wedge\tau$ for any $\tau$. But is that what we want? –  Joel David Hamkins Feb 1 at 14:36
    
@JoelDavidHamkins The issue seems true and shows that the definition is too board. –  user45939 Feb 2 at 8:39

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