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

156 concept calculus: much better thanDefinition Let (A, E, R, x) be a system. We say that c is a code for (A, E, R,x) using y,w,a 1 ,..., a 7 if and only if this holds where (A, E, R, x) is treatedas the 6-ary “relation” {:a∈ A ∧ E(b, c) ∧ R(d,e) ∧ f = x}.We say that c, y, w, a 1 ,...,a 7 is a full code for (A, E, R, x) if and only if c is acode for (A, E, R, x) using y,w,a 1 ,...,a 7 .Lemma 8.18 Let (A, E, R, x) be a system. Then the R maximal elements ofA are exactly the points E equivalent to x. Lety ∈ A. Then (A, E, R, x)|y is asystem.proof Let A, E, R, x, y be as given. It is obvious that z# is closed under E,Rpredecessors. Let B ={z: x/∈ z#\[x]}, where [x] is the E equivalence class of x.Then obviously [x] ⊆ B. Let x/∈ z#\[x], E(w, z). Then x/∈ w#\[x] given thatz# = w#. Let R(w, z). Then clearly w# ⊆ z#, by considering w# ∩ z#. Hence,w#\[x] ⊆ z#\[x], and so x/∈ w#\[x]. Hence, B contains [x] and is closed underE, R predecessors. So B = A.Thus, we have established that for all z in A, x /∈ z#\[x]. Therefore, x is Rmaximal. Now let C ={z ∈ A: ¬E(z, x) ∧ z is not R maximal}. Then C contains[x] and is closed under E, R predecessors. Hence, C = A. Therefore, every Rmaximal point is E equivalent to x.For the second claim, let y ∈ A. To see that (y#, E|y#, R|y#, y) is a system,it suffices to show that:i. Every C ⊆ y# that contains [y] and is closed under E|y#, R|y# predecessors isC. This is immediate from the definition of y#.ii. Every nonempty C ⊆ y# has an R|y# minimal element. Any R minimal elementis an R|y# minimal element.Definition We say that S is an isomorphism relation from (A, E, R, x) onto(A ′ ,E ′ , R ′ , x ′ ) if and only if:i. (A, E, R, x) and (A ′ ,E ′ ,R ′ , x ′ ) are systems.ii. S is a binary “relation.”iii. S ⊆ A × A ′ .iv. E(a,b) ∧ E ′ (c, d) → (S(a,c) ↔ S(b, d)).v. S(a,b) ∧ S(a,c) → E ′ (b, c).vi. S(a,b) ∧ S(c, b) → E(a,c).vii. (∀a ∈ A)(∃b ∈ A ′ )(S(a, b)).viii. (∀b ∈ A ′ )(∃a ∈ A)(S(a,b)).ix. S(a,b) ∧ S(c, d) → (R(a,c) ↔ R ′ (b, d)).We write (A, E, R, x) ≈ (A ′ , E ′ , R ′ , x ′ ) if and only if there is an isomorphism relationfrom (A, E, R, x) onto (A ′ , E ′ , R ′ , x ′ ).Lemma 8.19 ≈ is an equivalence relation on systems. If S is an isomorphismfrom (A, E, R, x) onto (A ′ , E ′ , R ′ , x ′ ), then S −1 is an isomorphism from (A ′ , E ′ ,R ′ , x ′ ) onto (A, E, R, x).