Theories of Proof-Theoretic Strength Ψ (Γ_Ω +1)

Abstract

The purpose of this article is to present a range of theories with proof-theoretic ordinal ψ(Γ Ω+1 ). This ordinal parallels the ordinal of predicative analysis, Γ 0 , and our theories are parallel to classical theories of strength Γ 0 such as ID <ω , FP 0 , ATR 0 , Σ 1 1 -DC 0 + (SUB), and Σ 1 1 -AC 0 + (SUB). We also relate these theories to the unfolding of ID 1 which was already presented in the PhD thesis of the first author as a system of strength ψ(Γ Ω+1 ).