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Article

Local reflection, definable elements and 1-provability

Archive for Mathematical Logic. 2020. Vol. 59. P. 979-996.

In this note we study several topics related to the schema of local reflection ๐–ฑ๐–ฟ๐—‡(๐‘‡) and its partial and relativized variants. Firstly, we introduce the principle of uniform reflection with ๐›ด๐‘›-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 ๐›ด๐‘›+2-conservativity of uniform ๐›ด๐‘›+1-reflection over relativized local ๐›ด๐‘›+1-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 ๐–ฑ๐–ฟ๐—‡(๐‘†), and its generalized versions. We relate this assertion to the uniform ๐›ด2-reflection schema and, in particular, obtain an alternative axiomatization of ๐–จ๐›ด1.