Some observations on Truth Hierarchies

We provide a precise representation for, and a calculation of the length of, possible path independent determinateness hierarchies of Field’s construction in [4]. We further show in the hierarchies Fα of Fieldian truth sets, and Herzberger’s Hα revision sequence starting from any recursive hypothesis for F0 (or H0) that essentially each Hα (or Fα) carries within it the history of the whole prior revision process; we use this to prove a result on the non-decreasing nature of such sequences. We demonstrate the existence of generalised liar sentences, that can be considered as diagonalising past the determinateness hierarchies definable in Field’s recent models. The ‘defectiveness’ of such diagonal sentences necessarily cannot be classified by any of the determinateness predicates of the model. They are ‘ineffable liars.’