Journal of the
Korean Mathematical Society JKMS

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 $\thomom$, the small Veblen ordinal, which is the proof-theoretic ordinal of the theory $\acaobi$. 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 $\acaobi$.

Keywords: independence results, Peano arithmetic, Kruskal's theorem

MSC numbers: 03F03, 03B20