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FEFERMAN’S COMPLETENESS THEOREM

Part of: Model theory Proof theory and constructive mathematics

Published online by Cambridge University Press:  10 January 2025

FEDOR PAKHOMOV Open the ORCID record for FEDOR PAKHOMOV [Opens in a new window] ,
MICHAEL RATHJEN  and
DINO ROSSEGGER Open the ORCID record for DINO ROSSEGGER [Opens in a new window]
FEDOR PAKHOMOV
Affiliation:
VAKGROEP WISKUNDE: ANALYSE LOGICA EN DISCRETE WISKUNDE UNIVERSITEIT GENT AND STEKLOV MATHEMATICAL INSTITUTE OF RUSSIAN ACADEMY OF SCIENCES GENT, BELGIUM AND MOSCOW, RUSSIA E-mail: fedor.pakhomov@ugent.be URL: https://homepage.mi-ras.ru/~pakhfn/
MICHAEL RATHJEN
Affiliation:
SCHOOL OF MATHEMATICS, UNIVERSITY OF LEEDS LEEDS, UK E-mail: m.rathjen@leeds.ac.uk URL: http://www1.maths.leeds.ac.uk/~rathjen/rathjen.html
DINO ROSSEGGER*
Affiliation:
INSTITUT FÜR DISKRETE MATHEMATIK UND GEOMETRIE TECHNISCHE UNIVERSITÄT WIEN, VIENNA, AUSTRIA URL: https://drossegger.github.io/
*
E-mail: dino.rossegger@tuwien.ac.at
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Abstract

Feferman proved in 1962 [6] that any arithmetical theorem is a consequence of a suitable transfinite iteration of full uniform reflection of $\mathsf {PA}$. This result is commonly known as Feferman’s completeness theorem. The purpose of this paper is twofold. On the one hand this is an expository paper, giving two new proofs of Feferman’s completeness theorem that, we hope, shed light on this mysterious and often overlooked result. On the other hand, we combine one of our proofs with results from computable structure theory due to Ash and Knight to give sharp bounds on the order types of well-orders necessary to attain the completeness for levels of the arithmetical hierarchy.

Keywords

Fefermancompleteness theoremreflection principles

MSC classification

Primary: 03F15: Recursive ordinals and ordinal notations 03F30: First-order arithmetic and fragments 03C57: Effective and recursion-theoretic model theory

Information

Type
Article
Information
Bulletin of Symbolic Logic , Volume 31 , Issue 3 , September 2025 , pp. 462 - 487
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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