 This provides a means to select the correct axiom from among V = LΩ, V = LΩS, V = LΩ(∗), and their variants. One simply looks at the L(Vλ+1) of each model (where the embedding axiom holds) and checks to see which has the true analogue of the structure theory of L(ℝ) under the assumption of ADL(ℝ). It is already known that certain pieces of the structure theory cannot hold in LΩ. But it is open whether they can hold in LΩS. 