IANA mathematician, but I read "axiomatic system" broadly as including not just the axioms but also the logic in which they are based. My understanding is that a common interpretation is that ZFC axioms are a list of 10 strings of symbols, which only have some sort of meaning when you pick a logic that gives meaning to these symbols. But I think also that this particular understanding of what axioms are ("formalism") is just one way of understanding truth in mathematics, and there are others. https://en.wikipedia.org/wiki/Philosophy_of_mathematics
As for this particular issue I think this wikipedia page is relevant: https://en.wikipedia.org/wiki/Impredicativity