In ordinal notations such as Stegert's (Ordinal Proof Theory of Kripke–Platek Set Theory Augmented by Strong Reflection Principles) and Rathjen's (An Ordinal Analysis of parameter free $\Pi_2^1$-Comprehension,) the reflection/projection instances utilize expressions of the form $\mathsf{M}_{\mathbb{M}}^{< \xi}\text-\mathsf{P}_m$ (or something similar) and I have heard them colloquially called "$\mathsf M\text-\mathsf P$-expressions". They seem interesting and powerful, yet I do not understand their purpose: all I managed to tell is that, in Stegert's notation, the $m$ in $\mathsf{P}_m$ corresponds to the "degree of $\Pi^1_m$-indescribable on" but I can't tell much more. Could anyone help?
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