Publisher Summary This chapter discusses the non-finitary procedures while retaining a purely formal point of view. In this manner, an extended version of metamathematics is obtained, which is called “syntax.” Within the framework of syntax, the chapter proves a stronger version of the completeness theorem for sentential logic. The syntax of elementary logic and of similar systems can be given an axiomatic basis in very much the same manner. The chapter illustrates a simple proof of the completeness theorem for elementary logic. An intermediate system—reduced logic—is introduced. The expressions of reduced logic are those formulas that can be obtained from expressions of elementary logic by replacing every free variable by some numeral. The chapter also discusses arithmetization of the syntax of sentential logic.