Semantic consistency of a theory T is defined by there being a model M in which all theorems of T are true. Syntactic consistency of a theory T is defined by there being no formulas A such that both A and it's negation ~A are provable. So let's assume semantic consistency and then assume syntactic inconsistency. Since T is inconsistent, both A and its negation ~A are provable. But since it's also semantically consistent, there must be a model where all of its theorems are true. So there must be a model M where both A and ~A is true, which is not possible. So semantic consistency implies syntactic consistency.
Is this conclusion right or am I severely misunderstanding something? I thought that semantic consistency only implied syntactic consistency under the additional assumption of soundness, but soundness was not assumed here. Is this conclusion a consequence of defining semantic consistency too broadly? Should it be limited to there being a model where the axioms are true?
