Università Roma Tre

Part 1: Some preliminary notions.
Order relations and trees, inductive definitions, proofs by induction, axiom of choice and Kőnig's lemma.

Part 2: Provability and satisyability.
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.

Part 3: 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. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 1 Dimostrazioni e modelli al primo ordine, Springer, 2014
https://sites.google.com/view/lm410/home

Proof of the compactness theorem for languages of any cardinality. Languages with equality. The compactness theorem for languages with equality. Correctness and completeness for languages with equality. L"owenheim-Skolem's theorem for (denumerable) languages with equality. The limits of the expressive power of first order languages. Elementary equivalence, substructures, elementary substructures. Isomorphsims and elementary equivalence. The notion of substructure. Elementary substructures and diagrams. The preservation theorems. Generalisations of the L"owenheim-Skolem's theorem. Completeness of a theory.

V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 1 Dimostrazioni e modelli al primo ordine, Springer, 2014