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Large cardinals such as weakly compact cardinals, measurable cardinals, strongly compact cardinals, and extendible cardinals all can be characterized in terms of a certain compactness theorem of certain strength.

For instance, weakly compact cardinals are precisely the cardinals $\kappa$ where the $\kappa$-compactness theorem holds in $L_{\kappa,\kappa}$ when we bound the number of non-logical symbols by $\kappa$. Strongly compact cardinals are precisely the cardinals where the $\kappa$-compactness theorem holds for $L_{\kappa,\kappa}$. Extendible cardinals are the cardinals where the compactness theorem holds for higher order infinitary logic. Furthermore, there are characterizations of at least weakly compact, measurable, and strongly compact cardinals involving different forms of compactness besides logical compactness.

Is there is a characterization of supercompact cardinals in terms of some sort of compactness theorem? Can any other large cardinal axioms besides the ones already mentioned be characterized in terms of some sort of compactness theorem?

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    $\begingroup$ Perhaps $\aleph_1$, viewed as a kind of large cardinal, can be characterized via compactness? Or the cardinals with uncountable cofinality? $\endgroup$
    – Joel David Hamkins
    Commented Mar 28, 2014 at 22:03
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    $\begingroup$ Not exactly compactness, but a possibly related notion, the least supercompact cardinal is the Lowenheim-Skolem-Tarski number of second-order logic. (This is due to Magidor and Vannanen.) $\endgroup$
    – Asaf Karagila ♦
    Commented Mar 28, 2014 at 22:43
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    $\begingroup$ @Noah: Actually, looking through the reference, I believe this is actually just by Magidor from somewhere in the 1970's. But he and Vannanen did some interesting work to extend it. See logic.math.helsinki.fi/people/jouko.vaananen/JV96.pdf $\endgroup$
    – Asaf Karagila ♦
    Commented Mar 28, 2014 at 23:18
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    $\begingroup$ @ Asaf Karagila: I guess you refer by "Vannanen" to Jouko Väänänen? $\endgroup$
    – TaQ
    Commented Mar 29, 2014 at 0:37
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    $\begingroup$ Even though Mohammad Golshani has given an affirmative answer that I accepted, I conjecture that there are other characterizations of supercompact cardinals that involve some sort of compactness. In particular, I conjecture that there is some purely combinatorial "compactness" theorem that characterizes supercompact cardinals. $\endgroup$
    – Joseph Van Name
    Commented Mar 29, 2014 at 20:08

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The answer is yes. In the paper "Compactness for omitting of types" (Annals of Mathematical Logic, vol 14(1) (1978), pp. 39-56), Benda gives such a characterization. He proves the following:

Theorem. The following are equivalent:

1) $\kappa$ is supercompact,

2) If $T$ is a theory in $L_{\kappa, \kappa}$ and $\Sigma(x,y)$ is a type such that $\{ \Delta \in P_{\kappa}(\Sigma) : T+(\exists x)\alpha_\Delta(x)$ has a model$ \}$ is in $ F_{\kappa}(\Sigma),$ then $T+(\exists x)\alpha_\Sigma(x)$ has a model.

Here $\alpha_\Delta(x)$ is the formula $\bigwedge(\exists y)\Delta \wedge (\exists y)\bigwedge\Delta $ and $F_{\kappa}(\Sigma)$ is the filter of club subsets of $P_{\kappa}(\Sigma)$

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  • $\begingroup$ Does there exist an analogous characterization for extendible cardinals, i.e. that references $L_{\kappa,\kappa}$? rather than $L^2_{\kappa}$ as in Magidor's paper "On the Role of Supercompact and Extendible Cardinals in Logic"? $\endgroup$
    – Thomas Benjamin
    Commented Mar 16, 2015 at 0:37
  • $\begingroup$ @ThomasBenjamin I have no idea, but maybe check to see if it is possible to extend the above proof. $\endgroup$
    – Mohammad Golshani
    Commented Mar 16, 2015 at 4:04
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Now I see you are also asking for combinatorial forms of compactness. I am not sure whether the tree property should be considered as a form of "compactness", but there are results by Jech, Di Prisco, Magidor, Zwicker and perhaps others which give characterizations of large cardinals in terms of variations of the tree property, and recent interest in such matters (together with new results!) has originated with Christoph Weiß' Dissertation of 2010, Subtle and Ineffable Tree Properties.

Similar to the case of the classical tree property, such stronger tree properties characterize large cardinals only under the assumption of (strong) inaccessibility. Without inaccessibility these tree properties can hold on small cardinals, too.

Besides Weiß' Dissertation, further results of this kind appear, for example, in Weiß, The combinatorial essence of supercompactness, Ann. Pure Appl. Logic 163 (2012), Viale and Weiß, On the consistency strength of the proper forcing axiom, Adv. Math. 228 (2011), no. 5, 2672-2687, Fontanella, Strong tree properties for two successive cardinals, Arch. Math. Logic 51 (2012), no. 5-6, 601-620. References to former results can be found in the above works, and also in the MR review of Laura's paper, MR2945570 by Andrew D. Brooke-Taylor.

At present nothing similar comes to my mind for larger cardinals, say, huge ones.

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    $\begingroup$ I believe it's unlikely that cardinals above Vopenka's principle in consistency strength will have logical characterizations, since Vopenka's principle already implies that every logic (with set-sized occurrence number) has a Hanf number, strong compactness number, Lowenheim-Skolem-Tarski number, etc. $\endgroup$
    – Keith Millar
    Commented Dec 8, 2019 at 18:52
  • $\begingroup$ In a sense you are right and Vopenka's principle furnishes the ultimate compactness theorem. On the other hand, the existence of some ultrafilter can be considered as a compactness property, since you can contruct a new model from existing ones, taking their ultraproduct. I am pretty sure that, say, huge cardinals have influences on model theory, maybe not of extreme interest, but they should have influence! $\endgroup$
    – Paolo Lipparini
    Commented Dec 15, 2019 at 23:31
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    $\begingroup$ Not really, but they do have interested when bringing in types (though not compactness in the usual sense). The collection of logics with set-sized occurrence number is an extremely large collection, and so pretty much any compactness theorem no matter how you express it will be bounded in consistency strength by VP. Keep in mind that also, strong compactness is itself an "ultimate theorem" for various logical properties; for example, assuming strong compactness of a logic $L$ at $\kappa$, every model of size at least $\kappa$ has arbitrarily large $L$-elementary extensions. $\endgroup$
    – Keith Millar
    Commented Dec 16, 2019 at 6:37

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