Notes on Wajsberg’s Proof of the Separation Theorem

Abstract

In his well-known paper M Wajsberg (1938) stated some important results on the intuitionistic propositional logic. But in connection with Wajsberg’s proof of the Separation Theorem (which was first formulated and proved by him), A. Church (1966) indicates that this paper of Wajsberg’s contains an error which is difficult to correct (see the correction of footnote 211). Further Church notes that the correct Gentzen-style proof of this theorem for intuitionistic predicate logic was given by H. B. Curry (1939) which was in print when Wajsberg’s paper appeared. Afterwards this proof was reproduced by S. C. Kleene (1952). Church writes also that for Curry’s proof essential is the Gentzen’s Cut Theorem but not the use of sequents which can be eliminated. Therefore the Cut Theorem can be applied in a suitable form to Hilbert type formulations (see H. B. Curry (1939) and K. Schütte (1950)). Concerning Wajsberg’s proof of the Separation Theorem Curry (1963) writes that he has never examined this proof, but judging by Bernays and an errata sheet to the book of Church (1956) (see footnote 211), the proof contains an error (p.250).