Abstract

In this paper, we introduce provability multilattice logic PMLn and multilattice arithmetic MPAn which extends first-order multilattice logic with equality by multilattice versions of Peano axioms. We show that PMLn has the provability interpretation with respect to MPAn and prove the arithmetic completeness theorem for it. We formulate PMLn in the form of a nested sequent calculus and show that cut is admissible in it. We introduce the notion of a provability multilattice and develop algebraic semantics for PMLn on its basis, by the method of Lindenbaum-Tarski algebras we prove the algebraic completeness theorem. We present Kripke semantics for PMLn and establish the Kripke completeness theorem via syntactical and semantic embeddings from PMLn into GL and vice versa. Last but not least, the decidability of PMLn is shown.