In The Theory of Parsing, Translation, and Compiling, Volume I, Section 0.2.1 (p.15 / 1972), Aho and Ullman casually write that [a]n alphabet need not be finite or even countable, but for all practical applications alphabets will be finite.However, in Introduction to Automata Theory, Languages, and Computation, 3rd Edition, Section 1.5.1 (p.28 / 2007), Hopcroft, Motwani and Ullman state that [a]n alphabet is a finite, nonempty set of symbols. Further, Sipser define[s] an alphabet to be any nonempty finite set. (Introduction to the Theory of Computation, 3rd Edition, Section 0.2 (p.13 / 2013).)Is it correct that an alphabet may be infinite, even uncountably so? If so, are there any resources that discuss infinite alphabets?
In the context of regular languages, must the alphabet be finite?
fl.formal languages;computability;automata theory;regular language;set theory