Arithmetical Foundations Recursion. Evaluation. Consistency

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Author(s) : Michael Pfender

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 8-2014

MSC 2000

03G30 Categorical logic, topoi
03B30 Foundations of classical theories
03D75 Abstract and axiomatic computability and recursion theory

Abstract :
Recursive maps, nowadays called primitive recursive maps, p. r. maps, have been introduced by Gödel in his 1931 article for the arithmetisation, gödelisation, of metamathematics. For construction of his undecidable formula he introduces a nonconstructive, non-recursive predicate beweisbar, provable. Staying within the area of (categorical) free-variables theory PR of primitive recursion or appropriate extensions opens the chance to avoid the two Gödel's incompleteness theorems: these are stated for Principia Mathematica und verwandte Systeme, "related systems" such as in particular Zermelo-Fraenkel set theory ZF and v. Neumann Gödel Bernays set theory NGB. On the basis of primitive recursion we consider mu-recursive maps as partial p. r. maps. Special terminating general recursive maps considered are complexity controlled iterations. Map code evaluation for PR is given in terms of such an iteration. We discuss iterative map code evaluation in direction of termination conditioned soundness, and based on this mu-recursive decision of primitive recursive predicates. This leads to consistency provability and soundness for classical, quanti ed arithmetical and set theories as well as for the p. r. descent theory piR, with unexpected consequences: We show inconsistency provability for the quanti ed theories, as well as consistency provability and logical soundness for the theory piR of primitive recursion, strengthened by an axiom scheme of non-infinite descent of complexity controlled iterations like (iterative) mapcode evaluation.

Keywords : primitive recursion, mu-recursion, code evaluation, complexity controlled iteration, soundness, decidability, Goedel theorems, inconsistency provability for set theory, constructive consistency