18. Large cardinals

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(ii) {λ < κ : λ is inaccessible} is stationary in κ. Proof. (i)⇒(ii): Let S = {λ < κ : λ is regular}, and S ′ = {λ < κ : λ is inaccessible}. Assume that κ is Mahlo. In particular, κ is uncountable and inaccessible. Suppose that C is club in κ. The set D = {λ < κ : λ is strong limit} is clearly club in κ too. If λ ∈ S ∩ C ∩ D, then λ is inaccessible, as desired. (ii)⇒(i): obvious. The following proposition answers a natural question one may ask after seeing Corollary 18.3. Proposition 18.5. Suppose that κ is minimum such that ι k = κ. Then κ is not Mahlo. Proof. Suppose to the contrary that κ is Mahlo, and let S = {λ < κ : λ is inaccessible} For each λ ∈ S, let f(λ) be the α < κ such that λ = ι α . Then α = f(λ) < λ by the minimality of κ. So f is regressive on the stationary set S, and hence there is an α < κ and a stationary subset S ′ of S such that f(λ) = α for all λ ∈ S ′ . But actually f is clearly a one-one function, contradiction. Mahlo cardinals are in a sense larger than “ordinary” inaccessibles. Namely, below every Mahlo cardinal κ there are κ inaccessibles. But now in principle one could enumerate all the Mahlo cardinals, and then apply the same idea used in going from regular cardinals to Mahlo cardinals in order to go from Mahlo cardinals to higher Mahlo cardinals. Thus we can make the definitions • κ is hyper-Mahlo iff κ is inaccessible and the set {λ < κ : λ is Mahlo} is stationary in κ. • κ is hyper-hyper-Mahlo iff κ is inaccessible and the set {λ < κ : λ is hyper-Mahlo} is stationary in κ. Of course one can continue in this vein. Weakly compact cardinals • A cardinal κ is weakly compact iff κ > ω and κ → (κ, κ) 2 . There are several equivalent definitions of weak compactness. The one which justifies the name “compact” involves infinitary logic, and it will be discussed later. Right now we consider equivalent conditions involving trees and linear orderings. • A cardinal κ has the tree property iff every κ-tree has a chain of size κ. Equivalently, κ has the tree property iff there is no κ-Aronszajn tree. • A cardinal κ has the linear order property iff every linear order (L,
Assume the linear order property, and let (T,
Page 1: 18. Large cardinals The study, or u
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 and 12: Lemma 18.9. Let A be a set of infin
Page 13 and 14: Proof. Suppose not, and let X = {b
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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