M\"unchhausen provability
Abstract
By Solovay's celebrated completeness result on formal provability we know that the provability logic GL describes exactly all provable structural properties for any sound and strong enough arithmetical theory with a decidable axiomatisation. Japaridze generalised this result by considering a polymodal version GLP of GL with modalities [n] for each natural number n referring to ever increasing notions of provability. Modern treatments of GLP tend to interpret the [n] provability notion as "provable in a base theory T together with all true 0n formulas as oracles". In this paper we generalise this interpretation into the transfinite. In order to do so, a main difficulty to overcome is to generalise the syntactical characterisations of the oracle formulas of complexity 0n to the hyper-arithmetical hierarchy. The paper exploits the fact that provability is 01 complete and that similar results hold for stronger provability notions. As such, the oracle sentences to define provability at level α will recursively be taken to be consistency statements at lower levels: provability through provability whence the name of the paper. The paper proves soundness and completeness for the proposed interpretation for a wide class of theories; namely for any theory that can formalise the recursion described above and that has some further very natural properties. Some remarks are provided on how the recursion can be formalised into second order arithmetic and on lowering the proof-theoretical strength of these systems of second order arithmetic.