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MÜNCHHAUSEN PROVABILITY

Part of: Proof theory and constructive mathematics

Published online by Cambridge University Press:  10 June 2021

JOOST J. JOOSTEN
JOOST J. JOOSTEN*
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF BARCELONABARCELONA, SPAINE-mail:jjoosten@ub.eduURL:http://www.phil.uu.nl/~jjoosten
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Abstract

By Solovay’s celebrated completeness result [31] on formal provability we know that the provability logic ${\textbf {GL}}$ describes exactly all provable structural properties for any sound and strong enough arithmetical theory with a decidable axiomatisation. Japaridze generalised this result in [22] by considering a polymodal version ${\mathsf {GLP}}$ of ${\textbf {GL}}$ with modalities $[n]$ for each natural number n referring to ever increasing notions of provability. Modern treatments of ${\mathsf {GLP}}$ tend to interpret the $[n]$ provability notion as “provable in a base theory T together with all true $\Pi ^0_n$ 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 $\Pi ^0_n$ to the hyper-arithmetical hierarchy. The paper exploits the fact that provability is $\Sigma ^0_1$ complete and that similar results hold for stronger provability notions. As such, the oracle sentences to define provability at level $\alpha $ 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.

Keywords

provability logicsordinal analysissecond order arithmetictransfinite polymodal provability

MSC classification

Primary: 03F03: Proof theory, general
Secondary: 03F35: Second- and higher-order arithmetic and fragments 03F45: Provability logics and related algebras (e.g., diagonalizable algebras)
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 3 , September 2021 , pp. 1006 - 1034
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Copyright
© Association for Symbolic Logic 2021

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