18. Large cardinals

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S ∩ C is stationary in κ. Clearly still S ∩ C ∩ λ is non-stationary in λ for every regular λ < κ. So we may assume from the beginning that S is a set of infinite <strong>cardinals</strong>. Let 〈λ ξ : ξ < κ〉 be the strictly increasing enumeration of S. Let ⎧ ⎡ ⎤⎫ ⎨ T = ⎩ s : ∃ξ < κ ⎣s ∈ ∏ ⎬ λ η and s is one-one⎦ ⎭ . η
Proof. Suppose not, and let X = {b ∈ B : G(b) ≠ b}. Since we are assuming that X is a nonempty subclass of A, choose b ∈ X such that y ∈ A and yRb imply that y /∈ X. Then contradiction. G(b) = {G(y) : y ∈ A and yRb} = {G(y) : y ∈ B and yRb} = {y : y ∈ B and yRb} = {y : y ∈ B and y ∈ b} = {y : y ∈ b} = b, Lemma <strong>18.</strong>13. Let κ be weakly compact. Then for every U ⊆ V κ , the structure (V κ , ∈, U) has a transitive elementary extension (M, ∈, U ′ ) such that κ ∈ M. (This means that V κ ⊆ M and a sentence holds in the structure (V κ , ∈, U, x) x∈Vκ iff it holds in (M, ∈, U ′ , x) x∈Vκ .) Proof. Let Γ be the set of all L κκ -sentences true in the structure (V κ , ∈, U, x) x∈Vκ , together with the sentences c is an ordinal, α < c (for all α < κ), where c is a new individual constant. The language here clearly has κ many symbols. Every subset of Γ of size less than κ has a model; namely we can take (V κ , ∈, U, x, β) x∈Vκ , choosing β greater than each α appearing in the sentences of Γ. Hence by weak compactness, Γ has a model (M, E, W, k x , y) x∈Vκ . This model is well-founded, since the sentence ¬∃v 0 v 1 . . . [ ∧ (v n+1 ∈ v n ) n∈ω ] holds in (V κ , ∈, U, x) x∈Vκ , and hence in (M, E, W, k x , y) x∈Vκ . Note that k is an injection of V κ into M. Let F be a bijection from M\rng(k) onto {(V κ , u) : u ∈ M\rng(k)}. Then G def = k −1 ∪ F −1 is one-one, mapping M onto some set N such that V κ ⊆ N. We define, for x, z ∈ N, xE ′ z iff G −1 (x)EG −1 (z). Then G is an isomorphism from (M, E, W, k x , y) x∈Vκ onto N def = (N, E ′ , G[W], x, G(y)) x∈Vκ . Of course N is still well-founded. It is also extensional, since the extensionality axiom holds in (V κ , ∈) and hence in (M, E) and (N, E ′ ). Let H, P be the Mostowski collapse of (N, E ′ ). Thus P is a transitive set, and (1) H is an isomorphism from (N, E ′ ) onto (P, ∈). (2) ∀a, b ∈ N[aE ′ b ∈ V κ → a ∈ b]. 208
Page 1 and 2: 18. Large cardinals The study, or u
Page 3 and 4: Assume the linear order property, a
Page 5 and 6: Now let α < κ; we show that B has
Page 7 and 8: so since |B| = κ it follows that |
Page 9 and 10: By induction, |F α | ≤ κ for al
Page 11: Lemma 18.9. Let A be a set of infin
Page 15 and 16: Clearly (V κ , ∈, U) |= σ. Supp
Page 17 and 18: (i) F is an ultrafilter. (ii) F is
Page 19 and 20: {α < κ : f α (β) = 0} and {α <
Page 21 and 22: • I0 • huge • Vopěnka • ex
Page 23: {X ⊆ A : µ(X) = µ(A)} is a κ-c

18. Large cardinals

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