I don't know much about large cardinals; so, I want to get a feeling of the landscape. Hence this question.

Definition. Whenever $C$ is a unary predicate in the language of ZFC, let us call $C$ semiabsolute iff for all transitive models $M$ of ZFC and all $x \in M,$ if $\mathcal{P}(x) \in M$, then $C(x)$ iff $C^M(x).$

Question. Do large cardinal properties tend to be semiabsolute? What are some examples of large cardinal properties that are not?

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    $\begingroup$ No, they do not. Most large cardinal properties of a cardinal $\kappa$ we study nowadays require looking beyond $\mathcal P(\kappa)$. Some require looking at $\mathcal P^n(\kappa)$ for some $n$, some require going beyond. For supercompactness, no $\mathcal P^\alpha(\kappa)$ suffices. $\endgroup$ – Andrés E. Caicedo Dec 17 '14 at 15:35
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    $\begingroup$ That said, what I think is the right formulation of your question is whether most large cardinal axioms are $\Sigma_2$ assertions about the universe. Supercompactness is not, for instance. But we can find large cardinals assertions with a $\Sigma_2$ formulation and consistency strength beyond that of the cardinal under consideration. For example, rather than having $C(\kappa)$ be "$\kappa$ is supercompact", you can consider $\hat C(\kappa)$: "$V_\kappa\models\mathsf{ZFC}+$ there is a proper class of supercompact cardinals". $\endgroup$ – Andrés E. Caicedo Dec 17 '14 at 15:39

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