TITLE: Difference between consistency and satisfiability
QUESTION [8 upvotes]: If a set of formula is consistent, there exist a model in which every formula is true. This is only if the set is satisfiable.
But satisfiability is the fact that it can be true so what is the difference between the 2 notions ?

REPLY [9 votes]: Consistency is a syntactic property. It means that there is no proof of contradiction from your axioms.
Satisfiability is a semantic property. It means that there is a model of the axioms.
In first-order logic (as well as propositional logic) the two notions are equivalent because the logic is sound and complete. Meaning a satisfiable theory is consistent, and a consistent theory is satisfiable.
Other logics, however, are not so lucky to have both of these properties and the two notions separate. In fact, if we do not assume the axiom of choice, then it is consistent that there is a theory which is consistent but not satisfiable.