Local reflection, definable elements and 1-provability

Archive for Mathematical Logic:1-18 (forthcoming)

Abstract
In this note we study several topics related to the schema of local reflection \\) and its partial and relativized variants. Firstly, we introduce the principle of uniform reflection with \-definable parameters, establish its relationship with relativized local reflection principles and corresponding versions of induction with definable parameters. Using this schema we give a new model-theoretic proof of the \-conservativity of uniform \-reflection over relativized local \-reflection. We also study the proof-theoretic strength of Feferman’s theorem, i.e., the assertion of 1-provability in S of the local reflection schema \\), and its generalized versions. We relate this assertion to the uniform \-reflection schema and, in particular, obtain an alternative axiomatization of \.
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DOI 10.1007/s00153-020-00732-9
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References found in this work BETA

The Incompleteness Theorems.Craig Smorynski - 1977 - In Jon Barwise (ed.), Handbook of Mathematical Logic. North-Holland. pp. 821 -- 865.
Proof-Theoretic Analysis by Iterated Reflection.Lev D. Beklemishev - 2003 - Archive for Mathematical Logic 42 (6):515-552.
Transfinite Recursive Progressions of Axiomatic Theories.Solomon Feferman - 1962 - Journal of Symbolic Logic 27 (3):259-316.
Reflection Principles and Their Use for Establishing the Complexity of Axiomatic Systems.G. Kreisel & A. Lévy - 1968 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 14 (7-12):97-142.
Reflection Principles and Their Use for Establishing the Complexity of Axiomatic Systems.Georg Kreisel & Azriel Lévy - 1968 - Zeitschrift für Mathematische Logic Und Grundlagen der Mathematik 14 (1):97--142.

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