Been reading through some model theory and got to a section on constructing models from syntax and i have been presented with the following problem, sorry for the lack of solution i just have no idea where to even begin, any help would be appreciated!! Really want to get my head around model theory.
Lindenbaum’s Lemma If Σ is a finitely satisfiable set of Lsentences then there is a finitely satisfiable, deductively closed and complete set Σ* of L-sentences such that Σ ⊆ Σ*
Problem
Let L be a countable language, and enumerate the set of L-sentences as (ϕn)n∈N. Let Σ be a finitely satisfiable set of L-sentences, and show that at least one of Σ ∪ {ϕ0} or Σ ∪ {¬ϕ0} is finitely satisfiable. Then build on this idea to prove Lindenbaum’s lemma in the case that L is countable.