I write about the issues with cardinality within ZFC in the second half of my next article in a lot more detail. I'd be happy to hear your thoughts. Check it out when you get a chance!
When I say "primitive set" what I am referring to is that a primitive set is used to build another set (nested within). It's my own language for describing what is happening since there is not a lot of widely available literature on this problem specifically. Rational numbers are primitive to the Real numbers because Rational numbers are used to build the Real numbers.
The big issue is that we have confused cardinality (number of elements in a set) with information (all the things that can be known about the objects the set represents). Using more elements to describe the same objects only makes our mathematics more verbose. This is what I discuss in the linked article above.
I am not sure I answered your question fully so let me know if you'd like more clarification on what I have said. Thanks!
