User blog comment:P進大好きbot/List of common mistakes on formal logic appearing in googology/@comment-11227630-20181018005915

I don't think ordinals as large as cardinals are necessary in OCFs.

To analyse "KP with large ordinal axiom" or subsystems of second-order arithmatic, the least ordinal working as a "diagonalizer" can be as small as the admissible ordinal \(\omega^\text{CK}_1\). Then more admissibles, then recursively inaccessibles, then higher recursive inaccessibility, then recursive Mahloness, then \(\Pi_n\)-reflecting, then (+1)-stable, then higher (+β)-stable, then "α is (+α)-stable", then (+)-stable, inaccessibly-stable, Mahlo-stable, and doubly (+1)-stable, etc.

The existence of these ordinals is provable in ZFC, until some point, such as an ordinal \(\alpha\) that \(L_\alpha\models\text{ZFC}\). If an ordinal notation needs such points, we need other methods to prove its well-foundedness.

But below such points, why we need cardinals to prove well-foundedness? Why can't we use these ordinals?