Arithmetical Foundations Recursion. Evaluation. Consistency Excerpt

dc.contributor.authorPfender, Michael
dc.description.abstractRecursive 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.subject.ddc510 Mathematiken
dc.subject.othercategorical logic, topoien
dc.subject.otherfoundations of classical theoriesen
dc.subject.otherabstract and axiomatic computability and recursion theoryen
dc.titleArithmetical Foundations Recursion. Evaluation. Consistency Excerpten
dc.typeResearch Paperen
tub.affiliationFak. 2 Mathematik und Naturwissenschaften::Inst. Mathematikde
tub.affiliation.facultyFak. 2 Mathematik und Naturwissenschaftende
tub.affiliation.instituteInst. Mathematikde
tub.publisher.universityorinstitutionTechnische Universität Berlinen
tub.series.issuenumber2013, 32en
tub.series.namePreprint-Reihe des Instituts für Mathematik, Technische Universität Berlinen
tub.subject.msc200003G30 Categorical logic, topoien
tub.subject.msc200003B30 Foundations of classical theoriesen
tub.subject.msc200003D75 Abstract and axiomatic computability and recursion theoryen
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