I have been trying for some time to get a grip on how large Mahlo cardinals are, but am finding the definition rather unsuggestive.
Let $\kappa$ be the smallest Mahlo cardinal. By definition, the set of inaccessible cardinals smaller than $\kappa$ is stationary in $\kappa$. Using this, I can prove (I think) that the smallest 1inaccessible cardinal is strictly smaller than $\kappa$.
But can I do better than this ?
Is the set of 1inaccessible cardinals smaller than $\kappa$ also stationary ?
If so, can I somehow iterate the argument to prove that for every ordinal m < $\kappa$, the set of minaccessible cardinals < $\kappa$ is stationary in $\kappa$ ?
The last property would imply that $\kappa$ is the $\kappa$th minaccessible cardinal for every m < $\kappa$ and therefore, truly gigantic, so I am very interested to know if this is true.
Apologies in advance if this is too basic for MathOverflow. I haven't seen an explicit proof or disproof of this property anywhere.
1 Answer
Yes. Erin Carmody gives a good account of this in her dissertation.
 Erin Carmody, Force to change large cardinal strength, arXiv:1506.03432, 2015.
If you see the material leading up to her theorem 11, she first develops the degrees of of inaccessibility beyond $\alpha$inaccessible and into things such as hyperinaccessible, richly inaccessible, utterly inaccessible and so on, ultimately defining a formal language of terms in $\Omega$, such as $(\Omega^3\cdot 5+\Omega)$inaccessible. And then she proves that if $\kappa$ is Mahlo, then it is $t$inaccessible for any term $t$ in her language and indeed a stationary limit of such $t$inaccessible cardinals.
This hierarchy goes way beyond merely $\kappa$inaccessible, and so it seems to be the answer to your question.
One basic observation that will get you a long way is the following.
Theorem. If $\kappa$ is Mahlo, then for any $A\subset\kappa$ there is a stationary class $S$ of inaccessible cardinals $\gamma$ for which $\langle V_\gamma,\in,A\cap\gamma\rangle\prec\langle V_\kappa,\in,A\rangle$.
This is just because the class of ordinals like that is club, and so by Mahloness it meets the regular cardinals. Now, as $A$ gets thinner, these $\gamma$ show that there are increasing degrees of hyperinaccessibility realized below the Mahlo cardinal $\kappa$. Any expressible property of $\kappa$ in $V_\kappa$ will hold on a stationary set of $\gamma$ below $\kappa$.
The basic idea is due to Mahlo, I believe. In Erin's presentation, she shows that the $t$inaccessible cardinals form a set in a certain normal filter, which therefore form a stationary set.

$\begingroup$ Sure, it was my pleasure. Erin was my PhD student. $\endgroup$ Feb 8, 2018 at 1:08

1$\begingroup$ Erin's hierarchy: twitter.com/earth1light/status/866189919148015617 $\endgroup$ Feb 8, 2018 at 13:54