Goodstein sequences for prominent ordinals up to the ordinal ofΠ11−CA0

Abstract

We introduce strong Goodstein principles which are true but unprovable in strong impredicative theories like ID n .

Pure $Σ_2$-Elementarity beyond the Core

“…Among the mathematically appealing examples exemplifying mathematical incompleteness are the Paris-Harrington theorem [31], Goodstein sequences [23,26,50], Kruskal's theorem [27,39], its extension by Friedman [44], the graph minor theorem by Robertson and Seymour, see [21] and, for a general reference on so-called concrete mathematical incompleteness, Friedman's book [20]. Concrete incompleteness refers to natural mathematical theorems independent of significantly strong fragments of ZFC, Zermelo-Fraenkel set theory with the axiom of choice.…”