Godel's Second Incompleteness Theorem for Definable Theories

2016/02/07 by Seraji, Payam, Chao, Conden
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1602.02416

Abstract

It is proved that if T is a Σ_n+1 Definable theory which is Σ_n-sound and extends PA, then T can not prove the sentence Σ_n-sound(T) that expresses the Σ_n-soundness of T. Optimality of this result is showed by constructing a Σ_n+1-definable and Σ_n-1-sound theory extending PA such that Σ_n-sound(T) is T-provable. It is also proved that no R.E. arithmetical theory, evevn very weak theories which are not Σ₁-complete, can prove Σ₁-soundness of itself.

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