Rcf 2 Evaluation and Consistency Ε&c * Π O R

We construct here an iterative evaluation of all PR map codes: progress of this iteration is measured by descending complexity within “Ordinal” O : = N[ω] of polynomials in one indeterminate, ordered lexicographically. Non-infinit descent of such iterations is added as a mild additional axiom schema (πO) to Theory PRA = PR + (abstr) of Primitive Recursion with predicate abstraction, out of forgoing part RCF 1. This then gives (correct) on-termination of iterative evaluation of argumented deduction trees as well, for theories πOR = PRA + (πO). By means of this constructive evaluation the Main Theorem is proved, on Termination-conditioned (Inner) Soundness for Theories πOR, Ordinal O extending N[ω]. As a consequence we get Self-Consistency for these theories πOR, namely πOR-derivation of πOR’s own free-variable Consistency formula ConπOR = ConπOR(k) =def ¬ProvπOR(k, pfalseq ) : N → 2, k ∈ N free. Here PR predicate ProvT(k, u) says, for an arithmetical theory T : number k ∈ N is a T-Proof code proving internally T-formula code u : k is an arithmetised Proof for u in Gödel’s sense. As to expect from classical setting, Self-Consistency of πOR gives (unconditioned) Objective Soundness. Termination-Conditioned Soundness holds “already” for PRA, but it turns out that at least present derivation of Consistency from this conditioned Soundness depends on schema (πO) of non-infinit descent in Ordinal O : = N[ω]. 0 this is an excerpt of part 2 of a cycle on Recursive Categorical Foundations, Abstract, Summary, basic section on Iterative – onterminating – evaluation, and References. 0 Legend of LOGO: ε for Constructive evaluation, C for Self-Consistency to be derived for suitable theories πOR strengthening in a “mild” way the (categorical) Free-Variables Theory PRA of Primitive Recursion with predicate abstraction TU Berlin, Mathematik, [email protected] last revised February 19, 2009