User blog comment:Deedlit11/Ordinal Notations VI: Up to a weakly compact cardinal/@comment-10429372-20131026081009/@comment-1605058-20131026105958

The problem is, by using similar diagonalization over and over again, we won't get very far. For example, if we just keep adding single diagonalizators like O, O_2,... we will probably never get to O_O_O_..., let alone weakly compact cardinals.