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First note to the following well known theorems:‎‎

Theorem (1): ‎The ‎notion ‎of ‎"‎$‎‎x$ ‎is a strongly inaccessible cardinal‎" ‎is ‎first ‎order ‎expressible ‎and ‎‎$‎‎\Pi_{1}$‎.

Theorem (2):‎‎ ‎The ‎notion ‎of ‎"‎$‎‎x$ is a measurable cardinal‎" ‎is ‎first ‎order ‎expressible ‎but ‎not ‎‎$‎‎‎\Pi‎_{1}$‎.‎

Theorem (3): ‎The ‎notion ‎of ‎"‎$‎‎x$ is a Reinhardt cardinal‎" ‎is ‎not ‎first order expressible.‎ ‎

Now there are some questions here:‎ ‎

Question (1): ‎Are ‎larger ‎large ‎cardinals ‎more ‎complicated ‎in ‎first ‎order ‎expressibilty? ‎Is ‎there ‎any ‎exception?‎ ‎

Question (2): ‎Is ‎there a‎ ‎non ‎first ‎orde‎r expressible large cardinal weaker than Reindhardt cardinal?‎ ‎

Question (3): ‎What ‎is ‎the ‎largest $‎‎\Pi_{1}$ - ‎expressible ‎large ‎cardinal? ‎For ‎example the notions of being a ‎Mahlo ‎or ‎weakly ‎compact ‎cardinal ‎are ‎first ‎order ‎expressible ‎and‎ $‎‎\Pi_{1}$.

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  • $\begingroup$ Being weakly compact is not $\Pi_1$. There are transitive models $M \subseteq N$ where $\kappa$ is weakly compact in $N$ but not in $M$. It is a $\Pi_2$ property. $\endgroup$
    – Monroe Eskew
    Commented Sep 11, 2013 at 14:51

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For question (1), there are many exceptions. For example, being superstrong is $\Sigma_2$ expressible, since it is witnessed inside a sufficiently large $V_\theta$, but this is stronger than strong in consistency strength, and being strong is $\Pi_3$. Similarly, being supercompact up to an inaccessible is $\Sigma_2$, but stronger than supercompact in strength, while being supercompact is $\Pi_3$. One can make many similar examples of very strong $\Sigma_2$ properties, by asserting that a strong $\Pi_3$ property holds up to an inaccessible cardinal. This reduces the complexity of the assertion, but is stronger consistency-wise.

For question (2), I would suggest Vopěnka's principle as a commonly considered large cardinal concept that, if consistent, is not first-order expressible as a single assertion in the language of set theory. This axiom is typically formulated in second-order theories such as GBC.

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  • $\begingroup$ Thanks for your elegant answers. What about question (3)? $\endgroup$
    – user36136
    Commented Sep 11, 2013 at 13:01
  • $\begingroup$ I suppose one could take a look through Cantor's upper attic cantorsattic.info/Upper_attic. It seems that $\Pi_1$ doesn't get you very far past Mahloness. Of course, you can get to hyper-Mahlo and somewhat beyond, but I'm not sure that anything larger is $\Pi_1$. Meanwhile, consistency assertions are $\Pi^0_1$, such as $\text{Con}(\text{ZFC}+\text{proper class of supercompact cardinals})$, and these can have very high consistency strength. So if these count as "large cardinal properties", then we shouldn't expect a largest one. $\endgroup$
    – Joel David Hamkins
    Commented Sep 11, 2013 at 13:06
  • $\begingroup$ So there is no direct relevance between largeness of a large cardinal and its first order expressibility. Do you have any information about this kind of relevance in a different sense? $\endgroup$
    – user36136
    Commented Sep 11, 2013 at 13:13
  • $\begingroup$ I don't know if you'd count this as a large cardinal, but a very strong $\Pi_1$ property of $\kappa$ is there are stationary many huge cardinals below $\kappa$. $\endgroup$
    – Monroe Eskew
    Commented Sep 11, 2013 at 14:15
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    $\begingroup$ @MonroeEskew, is that really $\Pi_1$? It looks at least $\Pi_2$ to me, since to say that $\delta$ is huge is $\Sigma_2$. $\endgroup$
    – Joel David Hamkins
    Commented Sep 11, 2013 at 14:18

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