Concatenation as a basis for arithmetic

General syntax, the formal part of the general theory of signs, has as its basic operation the operation of concatenation , expressed by the connective ‘⌢’ and understood as follows : where x and y are any expressions, x ⌢ y is the expression formed by writing the expression x immediately followed by the expression y. E.g., where ‘alpha’ and ‘beta’ are understood as names of the respective signs ‘α’ and ‘β’, the syntactical expression ‘alpha⌢beta’ is a name of the expression ‘ αβ ’. Tarski and Hermes have presented axioms for concatenation, and definitions of derivative syntactical concepts. Hermes has also related concatenation theory to the arithmetic of natural numbers, constructing a model of the latter within the former. Conversely, Gödel's proof of the impossibility of a complete consistent systematization of arithmetic depended on constructing a model of concatenation theory within arithmetic.