do we know if claude's formalization is built on top of zfc and not zfc+extra?
zfc itself is not sufficient, you need some layers of extra concepts formalization to fit specific problem domain(e.g. zfc doesn't define even basic arithmetics), which also could have potential issues.
Within a given inference system, one can define concepts. This doesn’t add any axioms. It is, in essence, just a way to abbreviate things.
ok, you now added some unknown inference system in addition to zfc