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 1-inaccessible cardinal is strictly smaller than $\kappa$.
But can I do better than this ?
Is the set of 1-inaccessible 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 m-inaccessible cardinals < $\kappa$ is stationary in $\kappa$ ?
The last property would imply that $\kappa$ is the $\kappa$-th m-inaccessible 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 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.