FRIEDMAN-WEIERMANN STYLE INDEPENDENCE RESULTS BEYOND PEANO ARITHMETIC

Abstract

We expose a pattern for establishing Friedman-Weiermann style independence results according to which there are thresholds of provability of some parameterized variants of well-partial-ordering. For this purpose, we investigate an ordinal notation system for <TEX>${\vartheta}{\Omega}^{\omega}$</TEX>, the small Veblen ordinal, which is the proof-theoretic ordinal of the theory <TEX>$({\prod}{\frac{1}{2}}-BI)_0$</TEX>. We also show that it sometimes suffices to prove the independence w.r.t. PA in order to obtain the same kind of independence results w.r.t. a stronger theory such as <TEX>$({\prod}{\frac{1}{2}}-BI)_0$</TEX>.