RCF2: Evaluation and Consistency

We construct here an iterative evaluation of all PR map codes: progress of this iteration is measured by descending complexity within “Ordinal” O := N[\omega] of polynomials in one indeterminate, ordered lexicographically. Non-infinit descent of such iterations is added as a mild additional axiom schema (\pi_O) to Theory PR_A = 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 PR_A+(\pi_O). By means of this constructive evaluation the Main Theorem is proved, on Termination-conditioned (Inner) Soundness for such theories, Ordinal O extending N[\omega]. As a consequence we get Self-Consistency for these theories, namely derivation of its own free-variable Consistency formula. As to expect from classical setting, Self-Consistency gives (unconditioned) Objective Soundness. Termination-Conditioned Soundness holds “already” for PR_A, but it turns out that at least present derivation of Consistency from this conditioned Soundness depends on schema (\pi_O) of non-infinit descent in Ordinal O := \N[\omega].

💡 Research Summary

The paper investigates a constructive strengthening of primitive‑recursive arithmetic (PR) by introducing an iterative evaluation procedure for all PR map codes and by adding a mild additional axiom schema, denoted (π_O), that guarantees non‑infinit descent in a specially designed ordinal. The base theory, called PR_A, is PR enriched with a predicate‑abstraction rule (abstr). On top of PR_A the authors define an ordinal O := N