User blog comment:DontDrinkH20/Some Explanations of Certain Large Cardinals/@comment-11227630-20181019090840

1. About the history of "stage cardinals": they were firstly imagined by me, then were used by other people. After long time, I visited Taranovsky's page, and see the similarity between his Ω (in Degree of Reflection) and the intuitive stage cardinal. And more recently, I got the "largeness" of \(\Pi_n^1\) indescribability, then realized that the original assignment of stages is false.

2. No ordinal \(\lambda\) is \(\lambda+1\)-Mahlo, or \(\lambda^+\)-Mahlo.

3. "Mahlo cardinals appear in the lower forms of indescribability" - Mahlo cardinals are exactly \(\Pi_0^1\)-indescribable over \(\Pi_0^1\)-indescribable cardinals, where ordinal \(\alpha\) is \(\Pi_n^m\)-indescribable over \(A\) if for every \(\Pi_n^m\) formula \(\phi\), \(\forall R\in V_{\alpha+1}(\langle V_\alpha,\in,R\rangle\models\phi\rightarrow\exists\beta\in A\cap\alpha(\langle V_\beta,\in,R\cap V_\beta\rangle\models\phi))\).

4. "Indescribable cardinals are useful in OCFs" - For OCF use, I don't care whether the large cardinal property is combinatorial or indescribability. What is important is how rich the hierarchy of such property is.

For example, after weakly compact cardinals is inaccessible limit of weakly compact cardinals, after that is Mahlo limit of weakly compact cardinals, after that is weakly compact limit of weakly compact cardinals, after that is a Mahlo cardinal below which weakly compact cardinals are stationary, after that is weakly compact cardinal below which weakly compact cardinals are stationary. But what comes next? It should be a cardinal with "weakly compact property" over weakly compact cardinals, and it should be large enough to be a "weakly compact cardinal below which weakly compact cardinals are stationary", and such cardinals are stationary in it, and arbitrarily more to come. After some discussion, I see that the definition of \(\Pi_1^1\)-indescribability is "large" enough.

Another example is that the OCF by Michael Rathjen used a concept of "reducible cardinals". The hierarchy of reducible cardinals have very rich structures. Hierarchy of stable ordinals are not very large in actual size, but they also have very rich structure like the hierarchy of reducible cardinals.