Generalizing Goodstein's theorem and Cichon's independence proof

We generalize Goodstein’s theorem (Goodstein 1944) and Cichon’s independence proof (Cichon 1983) to ${\mathrm{\Pi }}_1^1-{\mathrm{C}\mathrm{A}}_0$ using results from (Wilken 2026). The method is generalizable to stronger notation systems that provide unique terms for ordinals and enjoy Bachmann property.