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

158 concept calculus: much better thanproof Let S,S ′ ,A,E,R, x, A ′ , E ′ , R ′ , x ′ , y ′ , z ′ be as given. Apply Lemma8.22 to (A, E, R, x)|x and (A, E, R, x)|x, obtaining S = S ′ . By Lemma 8.21,E(y ′ , z ′ ), S = S ′ .Lemma 8.24 Let (A, E, R, x) be a system. Then y ∈ A ∧¬E(y,x) ↔(∃z)(R(z, x) ∧ y ∈ z#).proof Let A, E, R, x be as given. For the forward direction, it suffices toshow that B = {y ∈ A: E(y,x) ∨ (∃z)(R(z, x) ∧ y ∈ z#)} contains [x] and isclosed under E,R predecessors. Closure under E predecessors is obvious. Nowlet R(u, y), y ∈ B. IfE(y,x), then R(u, x) ∧ u ∈ x#, and so u ∈ B. SupposeR(z, x) ∧ y ∈ z#. Then u ∈ z#, and so u ∈ B.Conversely, let R(z, x) ∧ y ∈ z#. Suppose E(y,x). Then R(z, y), y ∈ z#, z ∈z#, contradicting Lemma 8.18.Lemma 8.25 Suppose (∀a)(R(a,x) → (∃a ′ )(R ′ (a ′ , x ′ ) ∧ (A, E, R, x)|a ≈ (A ′ ,E ′ , R ′ , x ′ )|a ′ )), (∀a ′ )(R ′ (a ′ , x ′ ) → (∃a)(R(a, x) ∧ (A ′ , E ′ , R ′ , x ′ )|a ′ ≈ (A, E, R,x)|a)). Then (A, E, R, x) ≈ (A ′ , E ′ , R ′ , x ′ ).proof Let A, E, R, x, A ′ , E ′ , R ′ , x ′ be as given. By Lemma 8.23, the a ′ inthe first hypothesis is unique up to E ′ . For each a such that R(a,x), let T a bethe unique isomorphism from (A, E, R, x)|a onto (A ′ , E ′ , R ′ , x ′ )|a ′ , where a ′is uniquely determined up to E ′ . The various T a cohere, according to Lemma8.22. Let T be the union of the various T a . Let T ∗ be T extended by {{:E(x,y) ∧ E ′ (x ′ ,z)}}.By the coherence, T respects E,E ′ and is univalent up to E ′ . By the secondhypothesis, T is onto the a ′ with R ′ (a ′ , x ′ ).Note that dom(T ) is the union of the a#, R(a,x), which, by Lemma 8.24, isA\[x]. Similarly, rng(T ) = A ′ \[x ′ ]. Hence, T ∗ also respects E, E ′ , is univalentup to E ′ , and is onto A ′ .Suppose u, v ∈ A\[x]. Let T (u, u ′ ),T(v, v ′ ). We claim thatR(u, v) ↔ R ′ (u ′ ,v ′ ).To see this, let u ∈ a#, R(a,x), v ∈ b#, R(b, x). Suppose R(u, v). Then u ∈ b#,and so u, v ∈ dom(T b ). By coherence, T b (u, u ′ ), T b (v, v ′ ). Hence, R ′ (u ′ , v ′ ). Theconverse is analogous.We also claim thatR(u, x) ↔ R ′ (u ′ ,x ′ )R(x,v) ↔ R ′ (x ′ ,v ′ ).Suppose R(u, x). Let u ∗ be such that R ′ (u ∗ , x ′ ) and (A, E, R, x)|u ≈ (A ′ , E ′ ,R ′ , x ′ )|u ∗ . Then T u (u, u ∗ ), and so T (u, u ∗ ). By coherence, E ′ (u ′ ,u ∗ ). Hence,R(u ′ , x ′ ).Note that x /∈ a#, x ′ /∈ b#, making the second equivalence vacuously true.This is because if x ∈ a# ∨ x ′ ∈ b#, then x is a maximal element of a# ∨ x ′ isa maximal element of b#, violating Lemma 8.18. From this we see that T is anisomorphism relation from (A, E, R, x) onto (A ′ , E ′ , R ′ , x ′ ).