Syntax

in mathematical logic

The description and study of a formal axiomatic theory as a purely symbolic system (in contrast to semantics, which is concerned with the meaning and content of objects of the formal theory). The difference between syntax and semantics is particularly important in the foundations of mathematics, where one studies formal theories whose semantics are intuitively insufficiently clear. In this case, the description and investigation of the syntax of the formal theory can often be realized by much more reliable and intuitively convincing methods within a certain meta-theory, and thus can serve as a basis for and indirect explanation of essential features of the complicated semantics of the theory in question. For example, in axiomatic set theory, the well-known results of K. Gödel on the consistency of the axiom of choice can be regarded as the syntactical and finitistically-provable assertion that if the Zermelo–Fraenkel formal theory is consistent, then it remains so after adding the axiom of choice.

When one passes beyond the limits of the foundations of mathematics into proof theory, the difference between syntax and semantics is not so essential. One uses so-called semi-formal systems, where the notion of a deduction depends on certain semantic stipulations. Formal languages can be defined essentially set-theoretically, in terms of infinitely long formulas, etc. On the other hand, for formal languages with bounded expressibility, such as the language of combinatory logic or an algorithmic language, the semantics can be formulated precisely in purely syntactical terms within the language itself.

References

[1]	R. Carnap, "The logical syntax of language" , Kegan Paul, Trench & Truber (1937) (Translated from German)
[2]	A. Church, "Introduction to mathematical logic" , 1 , Princeton Univ. Press (1956)
[3]	S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)