I'm self-studying first-order logic as a foundation of mathematics and struggling to understand what's the point of the symbolic system. In first-order logic, a formula is a "logical consequence" of axioms if it is true in each model that satisfies the axioms. This definition of the terminology "logical consequence" looks very odd to me.

If a model is something in real physical world, truth is definite as a physical reality that can be in princple just observed. However, in my opinion, mathematics does not physically exist. It is a construction using natural "language". In my naive mind, "truth" in a mathematical model is nothing but a consequence of a "proof" based on the definitions of the related mathematical objects and "logic" in a natural language. If there is a proof of a statement in a natural language, the statement is true. If there is a proof on the negation of the statement in the natural language, the statement is false. **In this view, the only difference between a symbolic theory (e.g., first-order theory) and a mathematical model is underlying langauges.**

Of course, one can argue that mathematics is an abtraction of some part of the real world or motivated by real world, but anyway it should be finally expressed in a natural language. Otherwise we could not make much advance in mathematics.

If a symbolic language has sufficiently expressive power, any "mathematical" statements in a natural language could be appropriately expressed in the symbolic language and vice versa. So we don't need to distinguish the symbolic theory and a model. In this view, when we discuss mathematics, the distinction between syntax and semantic is meaningless, and defining "logical consequence" using truth in a model looks like a tautology.

However, in some reason, if we have to use a symbolic langauge which is not sufficiently expressive (or one may insist that symbolic langauges cannot be as expressive as natural languages inherently), the forementioned definition of "logical consequence" may make sense. In this case, the axioms expressed in the symbolic langauge cannot completely capture the essence of the intended model which is defined in a natural langauge. Only part of the intended model is captured by the symbolic theory and so there could be another model (again defined in the natural language) that satisfies the same axioms.

Does this view make sense? Any recommendation on materials to study would be also helpful.