Papers › Well-orders in the transfinite Japaridze algebra
Well-orders in the transfinite Japaridze algebra
David Fernández-Duque, Joost J. Joosten
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This paper studies the transfinite propositional provability logics _Λ and their corresponding algebras. These logics have for each ordinal ξ< Λ a modality α. We will focus on the closed fragment of _Λ (i.e., where no propositional variables occur) and \emph{worms} therein. Worms are iterated consistency expressions of the form ξₙ…ξ₁ ⊤. Beklemishev has defined well-orderings <_ξ on worms whose modalities are all at least ξ and presented a calculus to compute the respective order-types. In the current paper we present a generalization of the original <_ξ orderings and provide a calculus for the corresponding generalized order-types o_ξ. Our calculus is based on so-called {\em hyperations} which are transfinite iterations of normal functions. Finally, we give two different characterizations of those sequences of ordinals which are of the form _ξ (A) _(ξ∈) for some worm A. One of these characterizations is in terms of a second kind of transfinite iteration called {\em cohyperation.}
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