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1 results

A NOTE ON PREDICATIVE ORDINAL ANALYSIS I: ITERATED COMPREHENSION AND TRANSFINITE INDUCTION
SATO KENTARO
Journal:

The Journal of Symbolic Logic / Volume 84 / Issue 1 / March 2019

Published online by Cambridge University Press:

13 February 2019, pp. 226-265

Print publication:

March 2019
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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.