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

160 concept calculus: much better thanR(z, x). Then (A ∗ , E ∗ , R ∗ , x ∗ ) witnesses the interpretation of this instance ofSeparation.Lemma 8.29The interpretation of Pairing holds.proof Let (A, E, R, x), (A ′ , E ′ , R ′ , x ′ ) be systems. Let y>A,A ′ . Let P holdof the full codes of (A, E, R, x), (A ′ , E ′ , R ′ , x ′ ). Now apply Lemma 8.27. Thiscreates a witness for Pairing applied to these two systems.Lemma 8.30The interpretation of Union holds.proof Let (A, E, R, x) be a system. Let P hold of the full codes of the(A, E, R, x)|y such that for some z, R(z, x) ∧ R(y,z). Let A < y, andapply Lemma 8.27. This creates a system that witnesses Union applied to(A, E, R, x).Lemma 8.31 Let (A, E, R, x), (A ′ , E ′ , R ′ , x ′ ) be systems. Suppose that for allsystems (A ∗ , E ∗ , R ∗ , x ∗ ) ∈ ′ (A, E, R, x), we have (A ∗ , E ∗ , R ∗ , x ∗ ) ∈ ′ (A ′ , E ′ , R ′ ,x ′ ). Then (A, E, R, x) is isomorphic to some system whose domain is a “subset”of A ′ .proof Let A, E, R, x, A ′ , E ′ , R ′ , x ′ be as given. Let B be the “set” consistingof the y such that R ′ (y,x ′ ), and (A ′ , E ′ , R ′ , x ′ )|y is isomorphic to some (A ∗ , E ∗ ,R ∗ , x ∗ ) ∈ ′ (A, E, R, x). Let C be the “set” consisting of the b that lies in the fieldof some (A ′ , E ′ , R ′ , x ′ )|y, y ∈ B. Let u>A,E,R, x, A ′ , E ′ , R ′ , x ′ . Then (C ∪{x ′ },E ∗∗ , R ∗∗ , x ′ ) is the desired system, where E ∗∗ is E restricted to C ∪{z ′ },and R ∗∗ is the binary “relation” given byR ∗∗(c, d) ↔ (R ′ (c, d) ∧ c, d ∈ C) ∨ (c ∈ B ∧ d = x ′ ).By Lemma 8.25, (A, E, R, x) ≈ (C ∪{x ′ },E ∗∗ , R ∗∗ , x ′ ).Lemma 8.32The interpretation of Power Set holds.proof Let (A, E, R, x) be a system. By Lemma 8.30, it suffices to form asystem (A ′ , E ′ , R ′ , x ′ ) such that for all systems α whose domain is a “subset” ofA, wehaveα ∈ ′ (A ′ , E ′ , R ′ , x ′ ). This is clear by Lemma 8.31.The following will be used in Section 6.9.Lemma 8.33 The interpretation of Foundation holds. In fact, the interpretationof the schematic form of Foundation holds.proof Let (∃x)(ϕ), where ϕ is a formula in ∈, =. Let the variable x be interpretedas the system (A, E, R, x). Let z be R minimal such that the interpretationof ϕ holds at (A, E, R, x)|z. Note that the interpretation of ϕ involves only ≈and ∈ ′ . Hence, (A, E, R, x)|z witnesses foundation for (∃x)(ϕ).The following is not needed but is of interest.Lemma 8.34The interpretation of (∀x)(∃y)(x ⊆ y ∧ y is transitive) holds.proof Let (A, E, R, x) be given. Then (A, E, R ′ , x) is as required, whereR ′ (a,b) ↔ R(a,b) ∨ (a ∈ A ∧ b = x).