I just want to know if people in the complexity initiated the name 'class' because the term 'class' in Zermelo-Fraenkel set theory or not?
For me, I assume the answer is Yes; because it doesn't seem right that it was coincidence and we use all time set theories tools to prove inclusions. But what made me asked this question is that I never found a reference in Sipser and Barak-Arrora that 'class' came from ZF-set theory.
Class in ZF-set theory is a collections of sets that have some common things. For example: Class P has set of languages which all of them are bounded by polynomial of the size of the input. Note that set is different from Class, see here.