On the basis of Poincaré and Weyl’s view of predicativity as invariance, we develop an extensive framework for predicative, type-free first-order set theory in which $\Gamma _0$ and much bigger ordinals can be defined as von Neumann ordinals. This refutes the accepted view of $\Gamma _0$ as the “limit of predicativity”.

Goodstein’s principle is arguably the first purely number-theoretic statement known to be independent of Peano arithmetic. It involves sequences of natural numbers which at first appear to diverge, but eventually decrease to zero. These sequences are defined relative to a notation system based on exponentiation for the natural numbers. In this article, we provide a self-contained and modern analysis of Goodstein’s principle, obtaining some variations and improvements. We explore notions of optimality for notation systems and apply them to the classical (...) Goodstein process and to a weaker variant based on multiplication rather than exponentiation. In particular, we introduce the notion of base-change maximality, and show how it leads to far-reaching extensions of Goodstein’s result. We moreover show that by varying the initial base of the Goodstein process, one readily obtains independence results for each of the fragments $\mathsf {I}\Sigma _n$ of Peano arithmetic. (shrink)

Cook and Reckhow [5] pointed out that $\mathcal {N}\mathcal {P} \neq co\mathcal {N}\mathcal {P}$ iff there is no propositional proof system that admits polynomial size proofs of all tautologies. The theory of proof complexity generators aims at constructing sets of tautologies hard for strong and possibly for all proof systems. We focus on a conjecture from [16] in foundations of the theory that there is a proof complexity generator hard for all proof systems. This can be equivalently formulated (for p-time (...) generators) without a reference to proof complexity notions as follows:•There exists a p-time function g stretching each input by one bit such that its range $rng(g)$ intersects all infinite $\mathcal {N}\mathcal {P}$ sets.We consider several facets of this conjecture, including its links to bounded arithmetic (witnessing and independence results), to time-bounded Kolmogorov complexity, to feasible disjunction property of propositional proof systems and to complexity of proof search. We argue that a specific gadget generator from [18] is a good candidate for g. We define a new hardness property of generators, the $\bigvee $ -hardness, and show that one specific gadget generator is the $\bigvee $ -hardest (w.r.t. any sufficiently strong proof system). We define the class of feasibly infinite $\mathcal {N}\mathcal {P}$ sets and show, assuming a hypothesis from circuit complexity, that the conjecture holds for all feasibly infinite $\mathcal {N}\mathcal {P}$ sets. (shrink)