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Iterated local reflection versus iterated consistency

Authors
Journal
Annals of Pure and Applied Logic
0168-0072
Publisher
Elsevier
Publication Date
Volume
75
Identifiers
DOI: 10.1016/0168-0072(95)00007-4
Disciplines
  • Mathematics

Abstract

Abstract For “natural enough” systems of ordinal notation we show that α times iterated local reflection schema over a sufficiently strong arithmetic T proves the same Π 1 0-sentences as ω α times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly at ε-numbers. We also derive the following more general “mixed” formulas estimating the consistency strength of iterated local reflection: for all ordinals α ⩾ 1 and all β, ( T α ) β ≡ Π 1 0 T ω α ·(1 + β) , ( T β ) α ≡ Π 1 0 T β + ω α . Here T α stands for α times iterated local reflection over T, T β stands for β times iterated consistency, and ≡ Π 1 0 denotes (provable in T) mutual Π 1 0-conservativity. In an appendix to this paper we develop our notion of “natural enough” system of ordinal notation and show that such systems do exist for every recursive ordinal.

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