Well-orders in the transfinite Japaridze algebra

This article studies the transfinite propositional provability logics GLP(A) and their corresponding algebras. These logics have for each ordinal xi . We will focus on the closed fragment of GLP(Lambda) (i.e. where no propositional variables occur) and worms therein. Worms are iterated consistency expressions of the form ... inverted perpendicular. Beklemishev has defined well-orderings < (xi) on worms whose modalities are all at least xi and presented a calculus to compute the respective order-types. In the current article, we present a generalization of the original <(xi) orderings and provide a calculus for the corresponding generalized order-types o(xi). Our calculus is based on so-called hyperations which are transfinite iterations of normal functions. Finally, we give two different characterizations of those sequences of ordinals which are of the form (xi epsilon On) for some worm A. One of these characterizations is in terms of a second kind of transfinite iteration called cohyperation