I do not know much about logic but I was wondering if someone could explain to me the importance of first order language in logic. In particular, what different does it make if something is expressible in first order logic language or not?
Just to give an example of what I am trying to understand, for instance, I know that if an axiomatic system has a model then it is consistent, now, what happens if an axiomatic system is expressible in first order logic and what happens if it cannot be expressed in FoL.