Arithmetical Foundations Recursion. Evaluation. Consistency Excerpt

 dc.contributor.author Pfender, Michael dc.date.accessioned 2021-12-17T10:10:53Z dc.date.available 2021-12-17T10:10:53Z dc.date.issued 2013-12-09 dc.description.abstract Recursive maps, nowadays called primitive recursive maps, PR 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 (original) Gödel's incompleteness theorems: these are stated for Principia Mathematica und verwandte Systeme, "relatedsystems" 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 μ-recursive maps as partial p. r. maps. Special terminating general recursive maps considered are complexity controlled iterations. Map code evaluation is then given in terms of such an iteration. We discuss iterative p. r. map code evaluation versus termination conditioned soundness and based on this decidability of primitive recursive predicates. This leads to consistency provability and soundness for classical, quantified arithmetical and set theories as well as for the PR descent theory πR, with unexpected consequences: We show inconsistency provability for the quantified theories as well as consistency provability and logical soundness for the theory πR of primitive recursion, strengthened by an axiom scheme of noninfinite descent of complexity controlled iterations like (iterative) mapcode evaluation. en dc.identifier.issn 2197-8085 dc.identifier.uri https://depositonce.tu-berlin.de/handle/11303/15764 dc.identifier.uri http://dx.doi.org/10.14279/depositonce-14537 dc.language.iso en en dc.rights.uri http://rightsstatements.org/vocab/InC/1.0/ en dc.subject.ddc 510 Mathematik en dc.subject.other categorical logic, topoi en dc.subject.other foundations of classical theories en dc.subject.other abstract and axiomatic computability and recursion theory en dc.title Arithmetical Foundations Recursion. Evaluation. Consistency Excerpt en dc.type Research Paper en dc.type.version submittedVersion en tub.accessrights.dnb free en tub.affiliation Fak. 2 Mathematik und Naturwissenschaften::Inst. Mathematik de tub.affiliation.faculty Fak. 2 Mathematik und Naturwissenschaften de tub.affiliation.institute Inst. Mathematik de tub.publisher.universityorinstitution Technische Universität Berlin en tub.series.issuenumber 2013, 32 en tub.series.name Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin en tub.subject.msc2000 03G30 Categorical logic, topoi en tub.subject.msc2000 03B30 Foundations of classical theories en tub.subject.msc2000 03D75 Abstract and axiomatic computability and recursion theory en
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