The issue is the notion that set theory can be the ultimate foundation of mathematics. ZF set theory is not a good foundation because it says “too much” about sets. For examples, every set in ZF has to have a power set, which means you can never talk about any object that doesn’t have a power set, such as the class of all Rings, or the surreal numbers. As it turns out, quite a lot of objects in mathematics are too large to be sets.

Therefore the ultimate foundation must be something else, such as categories, classes and types. You can still have ZF set theory but not at the very bottom.