On maximum conservative extensions

Abstract

We investigate the set of Π¹₂ sentences which are Π¹₁ conservative over theories of second order arithmetic. We exhibit new elements of these sets and conclude that the sets are Π₂ complete. Along the way, we show that, over the theory RCA₀, induction for Σ_n formulas has essentially no consequences for Δ_n+1 formulas.