 On its ordinary English interpretation, this axiom says that every set has a power set—a set which contains all and only the subsets of the set with which we  began.[25]  On its model-theoretic interpretation, however, the axiom says something much weaker. For any X ∈ M, the axiom ensures that we can find a Y ∈ M which contains exactly those subsets of X which also live in M (so, Y = { Z | Z ⊆ X ∧ Z ∈ M }). But if X is infinite, then most of the subsets of X won't live in the domain of M (since, after all, there are 2ℵ0 subsets of X, while the domain of M is only countable). So, the Y generated by the model-theoretic interpretation of the power set axiom will be much smaller than the real power set of X (Fraenkel et al. 1984; Tennant and McCarty 1987; Shapiro 1991; Hallett 1994; Giaquinto 2002; Bays 2007a). 