User blog comment:Bubby3/Larger cardinals vs indescribable cardinals/@comment-5529393-20161127140807

Let me add that an indescribable cardinal does not immediately correspond to a particular recursive ordinal in and of itself. For example, in if you take the ordinal notation presented in  my Ordinal Notations VI page   and replace "K" with another suitable cardinal (say the smallest $$\Pi^3_4$$ - indescribable cardinal), while leaving the rest of the notation the same, the notation would go up to the exact same recursive ordinal. It's the details of the ordinal notation that determine its precise strength. The reason K was chosen to be the smallest weakly compact cardinal was because it was a convenient cardinal to use due to its properties. (for example, it is closed under the Mahlo operation)