Generalizations of gödel’s incompleteness theorems for ∑n-definable theories of arithmetic

Review of Symbolic Logic 10 (4):603-616 (2017)
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Abstract

It is well known that Gödel’s incompleteness theorems hold for ∑1-definable theories containing Peano arithmetic. We generalize Gödel’s incompleteness theorems for arithmetically definable theories. First, we prove that every ∑n+1-definable ∑n-sound theory is incomplete. Secondly, we generalize and improve Jeroslow and Hájek’s results. That is, we prove that every consistent theory having ∏n+1set of theorems has a true but unprovable ∏nsentence. Lastly, we prove that no ∑n+1-definable ∑n-sound theory can prove its own ∑n-soundness. These three results are generalizations of Rosser’s improvement of the first incompleteness theorem, Gödel’s first incompleteness theorem, and the second incompleteness theorem, respectively.

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References found in this work

Extensions of some theorems of gödel and church.Barkley Rosser - 1936 - Journal of Symbolic Logic 1 (3):87-91.
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.
Gödel theorems for non-constructive logics.Barkley Rosser - 1937 - Journal of Symbolic Logic 2 (3):129-137.

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