For each $n > 0$, two alternative axiomatizations of the theory of strings over $n$ alphabetic characters are presented. One class of axiomatizations derives from Tarski's system of the Wahrheitsbegriff and uses the $n$ characters and concatenation as primitives. The other class involves using $n$ character-prefixing operators as primitives and derives from Hermes' Semiotik. All underlying logics are second order. It is shown that, for each $n$, the two theories are synonymous in the sense of deBouvere. It is further shown that each member of one class is synonymous with each member of the other class; thus that all of the theories are synonymous with each other and with Peano arithmetic. Categoricity of Peano arithmetic then implies categoricity of each of the above theories.