Question

 (compactness) let x be a set of formulas. x is said to be a finitely satisfiable set (fss) if every y ⊆fin x is satisfiable. equivalently, x is an fss if there is no finite subset {α1 ,α2 ,...,αn } of x such that ¬(α1 ∧ α2 ∧ ...∧ αn ) is valid. (note that if x is an fss we are not promised a single valuation v which satisfies every finite subset of x . each finite subset could be satisfied by a different valuation). show that: (i) every fss can be extended to a maximal fss. (ii) if x is a maximal fss then: (a) for every formula α, α ∈ x iff ¬α /∈ x . (b) for all formulas α,β, (α ∨ β) ∈ x iff (α ∈ x or β ∈ x ).

Answer 1

X is said to be a finitely satisfiable set (FSS) if every Y ⊆fin X is satisfiable. Equivalently, X is an FSS if there is no finite subset {α1,α2,...,αn} of X such that ¬(α1 ∧ α2 ...