User blog comment:Alemagno12/Huge Ordinal Analysis I: Z 2/@comment-1605058-20180208115720

In the Takeuti's paper Consistency Proofs of Subsystems of Classical Analysis a system SINN is defined (which is essentialy equivalent to \(\Pi^1_1-CA\)) and in chapter 2, section 6 (page 319) assignment of ordinal diagrams is defined.

Note that the assignment is not given to formulas by themselves, but rather to sequences (expressions of the form \(\Gamma\to\Delta\), with \(Gamma,\Delta\) fintie sets of formulas) which appear in some proof-figure. All the necessary definitions are provided earlier in the paper.