The basic logic course in school gives the impression that logic has both the syntax and the semantics aspects. Recently, I wonder whether the syntax part still plays an essential role in the current studies. Below are some of my observations, I hope the idea from the community can make them more complete.
Model theory: Even though model theory is stated in the language of logic, it can be viewed as the study of local isomorphism (see Poizat's "A course in Model Theory"). The syntax part is therefore a natural (though might be uncomfortable for some) way to view the theory rather than a necessity.
Recursion theory: The object of study is the notion of computability in different context. If we believe in Church-Turing Thesis, then these concept are independent of the formalism chosen.
Set theory: The intimate relationship between large cardinal and determinacy perhaps can suggest that this is a universal phenomenon. Will this phenomenon disappear if we change the language of mathematics to, for example, category theory?
Proof theory: I know too little to say anything.
If the observation is true, is it justified to demand that Turing degrees, and large cardinals receive the same mathematical status as, for example, prime numbers?