(* File: FOL_Axiomatic.thy Author: Asta Halkjær From This work is a formalization of the soundness and completeness of an axiomatic system for first-order logic. The proof system is based on System Q1 by Smullyan and the completeness proof follows his textbook "First-Order Logic" (Springer-Verlag 1968). The completeness proof is in the Henkin style where a consistent set is extended to a maximal consistent set using Lindenbaum's construction and Henkin witnesses are added during the construction to ensure saturation as well. The resulting set is a Hintikka set which, by the model existence theorem, is satisfiable in the Herbrand universe. *) theory FOL_Axiomatic imports "HOL-Library.Countable" begin section ‹Syntax›