Università Roma Tre

Provability and satisfiability

First order formal language: alphabet, terms, formulas, sequents. Structures for first order languages: structures, terms and formulas with parameters in a structure, value of terms, formulas and sequents. The calculus of sequents for first order logic: Gentzen's LK. Derivable sequents and derivations. Correctness of the rules of LK. Canonical analysis and fundamental theorem: construction of the canonical analysis (with and without cuts) and proof of the fundamental theorem of the canonical analysis. Consequences of the fundamental theorem: completeness theorem, compactness theorem, eliminability of cuts, L"owenheim-Skolem's theorem.

Towards proof-theory: the cut-elimination theorem

The cut-elimination procedure. Definition of the elementary steps of cut-elimination. First proof strategy (big reduction steps). Second proof strategy (reversion of derivations). The complexity of the cut-elimination procedure (sketch). Some immediate consequences of the cut-elimination theorem.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 1- Dimostrazioni e modelli al primo ordine. Springer, (2014)