Universes over Frege structures

Unifying the Philosophy of Truth

“…This idea of Kahle's is based on the notion of so-called Frege structures, and he has published a series of papers on theories of Frege structures over applicative base theories (Kahle 1999(Kahle , 2001(Kahle , 2003(Kahle , 2009(Kahle , 2011. This idea of Kahle's is based on the notion of so-called Frege structures, and he has published a series of papers on theories of Frege structures over applicative base theories (Kahle 1999(Kahle , 2001(Kahle , 2003(Kahle , 2009(Kahle , 2011.…”

“…This was investigated by using an external levelling and by Kahle (2003) with an internal levelling. Iterating truth.…”

Autonomous progression and transfinite iteration of self-applicable truth

“…Given a system S, Ax$ represents the set of the axioms of S and Prvs is a canonical provability predicate of S. Let P L^ be pure classical predicate logic for the language S?^ (thus with equality); then Axpi_^ represents the set of logical axioms (for S?^). Our following formulation of VF_< is based on the simpler version of VF in [5] (or [14]) rather than the original one in [4], DEFINITION 36. Given a p.r.…”

“…Lower bound. Kahle [14] gave a syntactical embedding of IDi in VF. In the present subsection, we generalize it to an embedding of ID_< in VF^.…”

“…Meta-induction on sf; use V2_<, (13) and Proposition 37. H We introduce two notations which generalize Clos^((y) and y/' in [14]: LEMMA 40. Let sf{R, Q, x, y) be an inductive operator form and y/{x) and 9{x) be arbitrary 5?^-formulae possibly with parameters y and v. Then, PI_2>-< h ( V y X V^C l o s^;^.…”