Notes on some second-order systems of iterated inductive definitions and Π11-comprehensions and relevant subsystems of set theory

Abstract

Pohlers's ordinal analysis in his monograph [12] contains some flaws and thereby ends up with incorrect proof-theoretic ordinals of several systems. The present paper determines their correct proof-theoretic ordinals and also supplements [12] with the ordinal analysis of some other relevant impredicative systems.

Classical Determinate Truth I

“…Moreover, neither can the truth predicate of FS be defined in CD nor can the truth predicate of CD be defined in FS. This follows from general arguments by Fujimoto [14] (see also Halbach [21]). We will show in Part II that CD can be consistently closed under NEC and CONEC.…”

“…205] originally introduced KF as an instrument to explain 'what notions and principles one ought to accept if one accepts the basic notions and principles of the theory', the foundational question which had long been one of the central themes of Feferman's work. 14 As a theory of truth per se, Feferman later proposed another system DT. Again, each predicate is assumed to have a domain of significance and to be meaningfully applicable only to objects in that domain.…”

“…Feferman [10] explicitly refers to Russell. 14 However, Feferman [10, p. 205] expressed 'I always thought that the KF axioms were a bit artificial for that purpose' and abandoned KF as such an instrument in the end. In place of KF, he proposed a new notion of unfolding of a schematic system in place of KF for the purpose in question.…”

“…Fujimoto [14] observed that DT is identical with the system FKF + Cons of non-classical partial truth whose evaluation rule is given by the Aczel-Feferman evaluation schema. The Aczel-Feferman evaluation schema is the same as the weak Kleene evaluation schema, with the exception of the evaluation rule of the conditional.…”

“… is proof-theoretically equivalent to . is proof-theoretically equivalent to (see [36] for its definition and Fujimoto [15] for its proof-theoretic analysis), and thus its proof-theoretic ordinal (if appropriately defined) is . …”