The omega-rule interpretation of transfinite provability logic |
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Authors: | David Fernández-Duque Joost J Joosten |
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Institution: | 1. Department of Mathematics, Ghent University, Krijgslaan 281, B 9000 Gent, Belgium;2. Dept. Lògica, Història i Filosofia de la Ciència, Universitat de Barcelona, Montalegre 6, 08001 Barcelona, Catalonia, Spain |
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Abstract: | Given a computable ordinal Λ, the transfinite provability logic has for each a modality intended to represent a provability predicate within a chain of increasing strength. One possibility is to read as ? is provable in T using ω-rules of depth at most ξ, where T is a second-order theory extending .In this paper we will formalize such iterations of ω-rules in second-order arithmetic and show how it is a special case of what we call uniform provability predicates. Uniform provability predicates are similar to Ignatiev's strong provability predicates except that they can be iterated transfinitely. Finally, we show that is sound and complete for any uniform provability predicate. |
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Keywords: | 03F03 03F45 03B30 03B45 Provability logic Arithmetic interpretation Iterated provability |
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