Background: I have just started reading J. Donald Monk's "Introduction to Set Theory," and I would like to double-check my understanding of the following paragraph:
The fundamental idea in making logic precise is to fix on a formal language, in principle different from English, for which we describe various notions (like term, sentence, theorem, and so on) with complete precision, and no ambiguity as in ordinary languages. Even so, we want the possibility of expanding our language from time to time. Thus, in set theory, we start with a very limited language, having only the symbol $ \in $ in addition to logical symbols, but we soon introduce many other symbols like $ \subseteq $,$ \cap $, $\mathscr{S}$, and so on by definitions. Hence, we need to describe, in a precise way, not a single language, but a whole class of languages, which have, however, many essential properties in common.
Is it accurate that he implies that each time a new symbol is added to a formal language, a new language is created, and so we need to describe the syntax of the class of languages that "start" from that minimal "set" of symbols?
That seems obvious but it is worded in a way that makes me uncertain.
