A descending hierarchy of reflection principles

Abstract

This paper can best be viewed as a portrait in miniature of a fascinating structure: a. descending hierarchy of reflection principles. Ascending hierarchies of reflection principles are amply studied, e.g. in Feferman's great paper Transfinite Recursive Progressions of Axiomatic Theories (Feferman[l962]). The problem adressed in the study of ascending hierarchies is: what is the a priori implicit in mathematical reasoning? Ascending hierarchies can be viewed as developing by continuing reflection in idealised time.. Descending hierarchies, in contrast, rest on presumption: in To we claim to know the reflection principle for T1, in Tl we claim reflection for T2, etcetera. Thus theories in a descending hierarchy are thoroughly ungrounded. Here we reach my motivation for writing this paper, which is: to provide a case-study of ungroundedness of provability-statements.