A note on iterated consistency and infinite proofs
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Schmerl and Beklemishev’s work on iterated reflection achieves two aims: it introduces the important notion of \(\varPi ^0_1\)-ordinal, characterizing the \(\varPi ^0_1\)-theorems of a theory in terms of transfinite iterations of consistency; and it provides an innovative calculus to compute the \(\varPi ^0_1\)-ordinals for a range of theories. The present note demonstrates that these achievements are independent: we read off \(\varPi ^0_1\)-ordinals from a Schütte-style ordinal analysis via infinite proofs, in a direct and transparent way.
KeywordsIterated consistency Ordinal analysis \(\varPi ^0_1\)-ordinal Infinite proofs \(\omega \)-rule Cut elimination
Mathematics Subject Classification03F05 03F25 03F15 03B30 03F30
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- 5.Schmerl, U.R.: A fine structure generated by reflection formulas over primitive recursive arithmetic. In: Boffa, M., van Dalen, D., MacAloon, K. (eds.) Logic Colloquium ‘78, pp. 335–350. North Holland, New York (1979)Google Scholar