I have some doubts regarding this concept.
DEF1: Given a formal language L, an interpretation of L is a mechanism that allow the ascription of a truth value to every sentence of L. If E is a set of formulas of L, a model of E is an interpretation of L that makes true every sentence in E. If T is a formal system, a model of T is a model of the set of theorems in T.
DEF2: A formal system is semantically complete when all its valid formulas (that is, formulas that are true under every interpretation of the language of the system) are theorems.
DEF3: A formal system is semantically complete when formulas true in every model of the system are theorems.
The first definition is informal and needs to be formalized differently for each "type" of language under consideration. My question is about DEF2 and DEF3, which to me don't look equivalent.
My question is: which one (between DEF2 and DEF3) is the correct (or more accepted) definition of semantic completeness?
A bit of formalization should make more clear why I think it's not obvious that DEF2 and DEF3 are equivalent:
T = Formal System , L = Language of T , Σ = Sentences of L
I = Set of Possible Interpretations of L
K = Set of Theorems of T
For i ∈ I, S(i) = { s ∈ Σ | i(s) = true}
V = Set of Validities = {s ∈ Σ | i ∈ I ⇒ i(s) = true}
M = Set of Models = {i ∈ I | k ∈ K ⇒ i(k) = true}
DEF2 completeness says V = ∩{S(i) | i ∈ I} ⊆ K
DEF3 completeness says ∩{S(i) | i ∈ M} = ∩{S(i) | (i ∈ I) & (k ∈ K ⇒ i(k) = true)} ⊆ K
It looks like DEF2 takes the intersection of a bigger family of sets.