Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

interpretation of z in b + sde 159Definition We write (A, E, R, x) ∈ ′ (A ′ , E ′ , R ′ , x ′ ) if and only if (∃y)(R ′ (y,x ′ )∧ (A, E, R, x) ≈ (A ′ , E ′ , R ′ , x ′ )|y).Definition The interpretation of L(∈, =) is as follows. The sets are the systems(A, E, R, x). The interpretation of ∈ is ∈ ′ . The interpretation of = is ≈.We now show that the interpretations of the axioms of Z hold. Recall that we areworking in B + SDE. Hence, we are going to show that the interpretations of theaxioms of Z are provable in B + SDE.Lemma 8.26The interpretation of Extensionality holds.proof Straightforward, using Lemma 8.25.Lemma 8.27 Let P be a property of full codes of systems whose fields are