Say you have some collection of axioms and you find a set $X$ fulfilling them. And the definition of the set $X$ involves another concept, e.g. the real number, but only in a way which refers to the axiomatically given properties of the reals.
Naively $X$ is a model of the axioms. However, if I don't have a model of the reals themselves, then $X$ isn't really fully given w.r.t. the set theory (the core axioms I say I work with) after all. And so it's not a real model of the axioms withing my theory, is it?
(A motivation for the question, although this the following is not quite so concrete, is that I want to figure out the sense in which a subset of an only implicitly defined set can by called a model. If $X=\{x\in U|P(x)\}$ and I define a subset $X'=\{x\in X|Q(x)\}=\{x\in U|P(x)\land Q(x)\}$, then the elements of $X'$ fulfill all of $X$'s eleemts "axioms", namely $P$ (I think. Might be that there are structual concerns relating the different $x$'s, but say I've checked that). And so it's kind of like a smaller model. But I'm a little confused, firstly because I've already working within the set theory here and secondly because this is not quite constructive, I just cut stuff off via a proposition $Q$.)