18. Large cardinals

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Suppose that ∆ ∈ [Γ]
By induction, |F α | ≤ κ for all α ≤ κ, and F κ is the set of all formulas. (One uses that κ is inaccessible.) Expand L to L ′ by adjoining a set C of new individual constants, with |C| = κ. Let Θ be the set of all subformulas of the sentences in Γ. Let 〈ϕ α : α < κ〉 list all sentences of L ′ which are of the form ∃xψ α (x) and are obtained from a member of Θ by replacing variables by members of C. Here x is a one-one sequence of variables of length less than κ; say that x has length β α . Now we define a sequence 〈d α : α < κ〉; each d α will be a sequence of members of C of length less than κ. If d β has been defined for all β < α, then ⋃ β
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: so since |B| = κ it follows that |
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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