Viewing First-order Language from Chomsky's Hierarchy

First-order languages are usually referred as formal languages in usual logic textbooks, but generative rules generally defines formula,rather than shows how to construct formula.In this article we illustrate a new understanding of the process of the formation of first-order languages from the point of view of Chomsky formal grammars.As a first-order language is recursive,so it must be a recursively enumerable language, thus there is a formal grammar to generate it.We illustrates one which is also a context-free grammar, hence any first-order language is more than a recursively enumerable language but also a context-free language. Furthermore,it is proved that first-order languages can also be generated with regular grammars by making use of the Godel encoding.Thus we can make a stronger conclusion that any first-order language is also a regular language.