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?‎
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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 $‎‎\Pi\_{1}$.



Are larger large cardinals less expressible?
For question (1), there are many exceptions. For example, being [superstrong](http://cantorsattic.info/Superstrong#Superstrong_cardinal) 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](http://cantorsattic.info/Vopenka) 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.


