In the superstructure approach of nonstandard analysis, one builds the superstructure $V(X)$ as ($X$ being a set) :
- $V_0(X)=X$
- For $n$ in $\mathbb{N}$, $V_{n+1}(X)=V_n(X)\cup\mathcal{P}(V_n(X))$
- $V(X)=\cup_{n\in\mathbb{N}}V_n(X)$
In order to have a correspondance between basic set operations and formulas on $(V(X),\in)$, one has to suppose that $X$ is a base set, that is $\forall x\in X,x\cap V(X)=\emptyset \wedge x\neq\emptyset$. How can one, in practise, replace a set by a base set of the same cardinal ? Can we even work in a base set of the same cardinal as if it were the first set ?
You have to require the set to be a base-set to begin with. Probably what Chang and Keisler mean by their remark is that there is a base-set in every cardinality (but I am not sure). For example, if you consider a real number to be a Dedekind cut (i.e., pair of suitable subsets of the rational numbers), then you get a base set. This point of view is developed in these class notes (see chapter 15 there).