Well-orders in the transfinite Japaridze algebra

Well-orders in the transfinite Japaridze algebra

The logic GLP is a polymodal logic that has for each ordinal α an operator [α], whose intended interpretation is a provability predicate in a hierarchy of theories of increasing strength. Its corresponding algebra is called the (transfinite) Japaridze algebra. There are various natural orders in this algebra that are based on comparing consistency strength of its elements. In particular, for ea...

Well orders in the transfinite Japaridze algebra II: Turing progressions and their well-orders

This paper is a follow up to [9, 8] and studies the poly-modal provability logics GLPΛ and natural well-founded orders therein. For each ordinal Λ one can define a propositional provability logic GLPΛ that has for each α < Λ a modal operator [α] corresponding to provability and with a dual operator 〈α〉 corresponding to consistency. By GLP we denote that class size logic that has a modality for ...

On Transfinite Knuth-Bendix Orders

In this paper we discuss the recently introduced transfinite Knuth-Bendix orders. We prove that any such order with finite subterm coefficients and for a finite signature is equivalent to an order using ordinals below ω, that is, finite sequences of natural numbers of a fixed length. We show that this result does not hold when subterm coefficients are infinite. However, we prove that in this ge...

Higher-Order Algebra with Transfinite Types

Well - Quasi - Orders Christian

Based on Isabelle/HOL’s type class for preorders, we introduce a type class for well-quasi-orders (wqo) which is characterized by the absence of “bad” sequences (our proofs are along the lines of the proof of Nash-Williams [1], from which we also borrow terminology). Our main results are instantiations for the product type, the list type, and a type of finite trees, which (almost) directly foll...