The definition of a complexity class reads in various different sources roughly as "a collection/set/... of decision problems". My question is: Is every complexity class a set (in the sense of axiomatic set theory)? Or are there complexity classes which are proper classes (as the name might suggest)?
Thanks for any comments!
This is a good question. For complexity classes containing decision problems, the answer is that they are just sets. Note that a decision problem is precisely a language; that is, a set $L \subseteq \{0,1\}^{*}$. The functional analogues of these classes (e.g., $\textsf{FP}$, the complexity class containing functions which are polynomial-time computable) are also sets.
It may be possible that when one delves deeply into logic, there are complexity classes which are proper classes. I am not a logician, so I do not know. Nor have I ever heard of this happening from complexity theorists.