Sure, but we still have a “class”. “Classes” are either crude syntactic sugar for “formulae” (as in ZFC) or they’re a slightly more refined syntactic sugar for “formulae” (as in BGC). In either case, classes are ubiquitous—for instance, ordinal addition isn’t a function either, but we prove things about it just as if it was.
Sure, but we still have a “class”. “Classes” are either crude syntactic sugar for “formulae” (as in ZFC) or they’re a slightly more refined syntactic sugar for “formulae” (as in BGC). In either case, classes are ubiquitous—for instance, ordinal addition isn’t a function either, but we prove things about it just as if it was.