# why is first-order logic strongest?

I get how first-order logic has Lowenheim-Skolem, compactness theorem, but I am not sure why this leads to first-order logic being strongest. All Lowenheim-Skolem seems to say is that for first-order countable infinite theory, there exists countable infinite model. And all compactness theorem seems to say is that for any subset of the aforementioned theory, there must be a model. What are consequences of not having those?

• – Asaf Karagila Feb 21 '13 at 11:56
• Compactness and Loweheim-Skolem are very nice results. However, they could be viewed as indicating weakness rather than strength, since they are ways in which standard first-order logic has less expressive strength than second-order logic, or certain infinitary logics. – André Nicolas Feb 21 '13 at 12:38

One notion of "strength" of the logic has to do with which classes of structures appear as a class of models for some theory in the logic. So logic $L$ is stronger than or equivalent to logic $L'$ ($L\succeq L'$) if every time a class of structures is "elementary" for $L'$ (can be picked out as the class of models of some $L'$-theory), it is also "elementary" for $L$.