TITLE: Are larger large cardinals less expressible?
QUESTION [7 upvotes]: 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.‎ 
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Now there are some questions here:‎
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Question (1): ‎Are ‎larger ‎large ‎cardinals ‎more ‎complicated ‎in ‎first ‎order ‎expressibilty? ‎Is ‎there ‎any ‎exception?‎
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Question (2): ‎Is ‎there a‎ ‎non ‎first ‎orde‎r expressible large cardinal weaker than Reindhardt cardinal?‎
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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 $‎‎\Pi_{1}$.

REPLY [9 votes]: 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.