Well-orders in the transfinite Japaridze algebra
David Fernández–Duque, Joost J. Joosten · Logic Journal of IGPL · 2014
This article studies the transfinite propositional provability logics GLPΛ and their corresponding algebras. These logics have for each ordinal ξ < Λ a modality 〈ξ〉. We will focus on the closed fragment of GLPΛ (i.e. where no propositional variables occur) and 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 article, we present a generalization of the original < ξ orderings and provide a calculus for the corresponding generalized order-types oξ. 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 〈oξ(A)〉ξ ∊ On for some worm A. One of these characterizations is in terms of a second kind of transfinite iteration called cohyperation.