Local reflection, definable elements and 1-provability

Archive for Mathematical Logic:1-18 (forthcoming)

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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