Systems of Explicit Mathematics with Non-Constructive µ-Operator, Part II

Systems of explicit mathematics were introduced in Feferman [4]; these provide axiomatic theories of operations and classes for the abstract development and prooftheoretic analysis of a variety of constructive and semi-constructive approaches to mathematics. In particular, two such systems T0 and T1 were introduced there, related roughly to constructive and predicative mathematics, respectively. T1 is obtained from T0 by adding a single axiom for the non-constructive but predicatively acceptable quantification operator eN over the natural numbers. However, since T1 (like T0) contains an axiom IG for a general impredicative inductive generation operator, it actually goes far beyond the limits of predicativity as measured by the Feferman-Schütte ordinal Γ0. Much precise proof-theoretic information was subsequently obtained about T0 and various of its subsystems; cf. Feferman [7], the two chapters of Feferman and Sieg in [2], Jäger and Pohlers [16] and Jäger [14]. Corresponding work on subsystems of T1 has been slower to be achieved. The first was for a theory VT(μ) of variable types with non-constructive μ operator (interdefinable with eN) in Feferman [5], which may be considered to be a subtheory of T1 without the J (join) and IG axioms. A proof was sketched there of the proof-theoretic equivalence of VT(μ) with (Π∞-CA)<ε0 (corresponding to ramified or predicative analysis up to level εo), and of the equivalence of a subsystem Res-VT(μ) with Peano arithmetic PA, where in Res-VT(μ), induction is restricted to (abstractly) decidable sets. Improved versions of these systems with corresponding results due to the present authors were stated in Feferman [9], but without proofs.