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  • $\begingroup$ But Andreas, does this actually show non-linearity in the proposed order? I don't think so. After all, from the existence of both a supercompact and a huge, we can prove that the least huge is smaller than the least supercompact, and so Huge $\lt$ supercompact in this order. The point instead is that $\lt$ is not the right order, since it differs so strongly from the consistency strength order. A more extreme example would be that Huge $\lt$ $\Sigma_2$-reflecting, since $\Sigma_2$ reflecting cardinals will reflect hugeness, but this is a strong reversal of the consistency strength order. $\endgroup$
    – Joel David Hamkins
    Commented May 3, 2011 at 0:33
  • $\begingroup$ Similarly, Supercompact $\lt$ $\Sigma_3$-reflecting on the proposed order, even though $\Sigma_3$-reflecting cardinals are very weak in consistency strength (below Mahlo, or even just ZFC, if you drop inaccessibility requirement). $\endgroup$
    – Joel David Hamkins
    Commented May 3, 2011 at 0:35
  • $\begingroup$ @Joel: I didn't intend to imply that this is a nonlinearity in either of the orders (size or consistency strength). It's just a disagreement between the two orders. $\endgroup$
    – Andreas Blass
    Commented May 3, 2011 at 2:17
  • $\begingroup$ Does there or can there exist a large cardinal axiom A such that (1) A is local (2) "Supercompact"<A (3) A is stronger than "Supercompact" in consistency strength? $\endgroup$
    – Garabed Gulbenkian
    Commented May 13, 2011 at 18:18