A NOTE ON PREDICATIVE ORDINAL ANALYSIS I: ITERATED COMPREHENSION AND TRANSFINITE INDUCTION
Kentaro Sato · Journal of Symbolic Logic · 2019
Abstract We determine the proof-theoretic ordinals (i) of ${\cal C} - {\bf{TI}}[\alpha ]$ , the transfinite induction along α, for any hyperarithmetical level ${\cal C}$ , in the first order setting and (ii) of any combination of iterated arithmetical comprehension and ${\cal C} - {\bf{TI}}[\alpha ]$ for ${\cal C}\, \equiv \,{\rm{\Pi }}_k^i ,{\rm{\Sigma }}_k^i$ ( $i\, = \,0,1$ ) in the second order setting.