Abstract
In the 1990s Yankov published two papers containing important contributions to proof theory and based on the application of a dialogical interpretation of proofs. In both cases the method is used for providing constructive proofs of important metalogical results concerning classical logic and fundamental mathematical theories. In the first paper it is shown that impredicative extensions of intuitionistic versions of arithmetic, analysis and set theory, enriched with suitable bar induction schemata, are sufficiently strong for proving the consistency of their classical counterparts. In the second paper the same method is applied to provide a constructive proof of the completeness theorem for classical logic. In both cases a version of a one-sided sequent calculus in Schütte-style is used and cut elimination is established in the second case. Although the obtained results are important, and the applied method is original and interesting, they have not received the attention they deserve from the wider community of researchers in proof theory. In this paper we briefly recall the content of both papers. We focus on essential features of Yankov’s approach and provide comparisons with other results of similar character.
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Notes
- 1.
Nowadays the view of Tait is often accepted according to which finitism is identified with the quantifier-free system of primitive recursive arithmetic PRA.
- 2.
In fact, a similar system of this kind independently introduced by Jaśkowski (1934) was called ‘suppositional calculus’.
- 3.
More on the history and types of natural deduction and sequent calculi is in Indrzejczak (2010).
- 4.
In the meantime he proved in a purely finitist way the consistency of simple type theory (Gentzen 1936b) but without the axiom of infinity so it is a rather weak result.
- 5.
The second edition is preferred since it contains also valuable appendices written by leading researchers in the field.
- 6.
Such a result was already present in Gentzen (1934).
- 7.
In particular research concerning bounded arithmetic (Parikh, Buss), reverse mathematics or predicative analysis were not mentioned since they are not related to Yankov’s work.
- 8.
Illustrated with an anecdote about two chess players who have a simultaneous match with a chess master and successively repeat his previous moves.
- 9.
Classically the fan theorem is just the contrapositive of König’s lemma, but intuitionistically they are not equivalent.
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Indrzejczak, A. (2022). Dialogues and Proofs; Yankov’s Contribution to Proof Theory. In: Citkin, A., Vandoulakis, I.M. (eds) V.A. Yankov on Non-Classical Logics, History and Philosophy of Mathematics. Outstanding Contributions to Logic, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-031-06843-0_3
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