 Our next step is therefore to understand H(ω2). It actually turns out that we will be able to understand slightly more and this is somewhat technical. We will be concerned with the structure  ⟨H(ω2), ∈, INS, AG⟩ ⊧ φ, where INS is the non-stationary ideal on ω1 and AG is the interpretation of (the canonical representation of) a set of reals A in L(ℝ). The details will not be important and the reader is asked to just think of H(ω2) along with some “extra stuff” and not worry about the details concerning the extra stuff.[6] 