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  • $\begingroup$ Thanks for your elegant answers. What about question (3)? $\endgroup$ 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. $\endgroup$ 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? $\endgroup$ 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 $\kappa$. $\endgroup$ 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 $\Sigma_2$. $\endgroup$ Commented Sep 11, 2013 at 14:18