There is not one set theory, there are many.
The author doesn't like Zermelo–Fraenkel set theory which is fine. And semantics in ZF set theory is for some theories not possible or intuitive.
Semantics in category theory sets (Topos) is usually the way to go (1).
"Topoi behave much like the category of sets" (2)
Lawvere, a renowned category theorist, wrote a book "SETS FOR MATHEMATICS", which describes different kinds of sets (topoi).
(1) https://ncatlab.org/nlab/show/relation+between+type+theory+a...