Well-orders in the transfinite Japaridze algebra II: Turing progressions   and their well-orders
  We study transfinite extensions of Japaridze's provability logic GLP and the well-founded relations that naturally occur within them. Every ordinal induces a partial order over the class of "words," which are iterated consistency statements expressible within GLP. Well-ordered restrictions of these partial orders have been studied previously; in this paper we consider the unrestricted partial orders, which are no longer linear but remain well-founded. These unrestricted partial orders bear important repercussions on modal semantics for GLP and on Turing progressions. 
David Fern\'andez-Duque, Joost J. Joosten
math.LO