Abstract This volume is one of two special issues collecting articles by invited speakers of the conference ‘Celebrating 90 Years of Gödel’s Incompleteness Theorems’ held in Nürtingen (Germany) in July 2021. The conference was organized by the Carl Friedrich von Weizsäcker Center at the University of Tübingen with support by the ERC-funded project Gödel Enigma: Rediscovering Kurt Gödel through his unpublished works at the University of Helsinki and by the Kurt Gödel Society in Vienna.

A formulation of lattice theory as a system of rules added to sequent calculus is given. The analysis of proofs for the contraction-free calculus of classical predicate logic known as G3c extends to derivations with the mathematical rules of lattice theory. It is shown that minimal derivations of quantifier-free sequents enjoy a subterm property: all terms in such derivations are terms in the endsequent. An alternative formulation of lattice theory as a system of rules in natural deduction style is given, both with explicit meet and join constructions and as a relational theory with existence axioms. A subterm property for the latter extends the standard decidable classes of quantificational formulas of pure predicate calculus to lattice theory.

Abstract. A sequent calculus is given in which the management of weakening and contraction is organized as in natural deduction. The latter has no explicit weakening or contraction, but vacuous and multiple discharges in rules that discharge assumptions. A comparison to natural deduction is given through translation of derivations between the two systems. It is proved that if a cut formula is never principal in a derivation leading to the right premiss of cut, it is a subformula of the conclusion. Therefore it is sufficient to eliminate those cuts that correspond to detour and permutation conversions in natural deduction.