In formal languages, alphabet is the set of all symbols used to form words in our language.
Why is the notion of "set" used in this definition, instead of some other kind of collection, e.g. class? Does this sort of thing come from purely historical reasons or is there some deeper, perhaps philosophical or logical, justification? Does this somehow relate to proving that the alphabet we are describing exists, which seems to be easier in set theory?
Also, perhaps the following few thoughts might be related to my question.
ZFC axioms are designed to replicate the intuitive properties of sets, so listing them seems like a good way to define sets. Listing them in natural language such as English and defining first-order languages and then building a first-order ZFC theory isn't a problem, or so I thought. But I didn't pay care to my reasoning since I forgot the axiom schema of replacement, which uses the concept of first-order formulas, which in turn must be described by first order languages, so natural language listing of the axioms would result in circularity.
This axiom schema isn't needed to prove the existence of alphabet usually used in first-order language of set theory.
Since we are selecting and using symbols from the alphabet, the axiom of choice, which uses the concept of a WFF too, is needed when defining a formal language. So there is still no way not to get rid of circular results when defining sets using ZFC in natural languages though.
This seems like a pretty bad result. So why not use some other collection than set to define alphabet?