18. Large cardinals

In fact, suppose that a, b ∈ N and aE ′ b ∈ V κ . Let the individual constants used in the
expansion of (V κ , ∈, U) to (V κ , ∈, U, x) a∈Vκ be 〈c x : x ∈ V κ 〉. Then
(V κ , ∈, U, x) a∈Vκ |= ∀z
[
z ∈ k b → ∨ w∈b(z = k w )
]
,
and hence this sentence holds in (N, E ′ , G[W], x, G(y)) x∈Vκ as well, and so there is a w ∈ b
such that a = w, i.e., a ∈ b. So (2) holds.
(3) ∀a, b ∈ V κ [a ∈ b → aE ′ b]
In fact, suppose that a, b ∈ V κ and a ∈ b. Then the sentence k a ∈ k b holds in (V κ , ∈
, U, x) x∈Vκ , so it also holds in (N, E ′ , G[W], x, G(y)) x∈Vκ , so that aE ′ b.
We have now verified the hypotheses of Lemma 18.12. It follows that H ↾ V κ is
the identity. In particular, V κ ⊆ P. Now take any sentence σ in the language of (V κ , ∈
, U, x) x∈Vκ . Then
(V κ , ∈, U, x) x∈Vκ |= σ iff (M, E, W, k x ) x∈Vκ |= σ
iff
(N, E ′ , G[W], x) x∈Vκ |= σ
iff (P, ∈, H[G[W]], x) x∈Vκ |= σ.
Thus (P, ∈, H[G[W]]) is an elementary extension of (V κ , ∈, U).
Now for α < κ we have
(M, E, W, k x , y) x∈Vκ |= [y is an ordinal and k α Ey],
hence
(N, E ′ , G[W], x, G(y)) x∈Vκ |= [G(y) is an ordinal and αE ′ G(y)],
hence
(P, ∈, H[G[W]], x, H(G(y))) x∈Vκ |= [H(G(y)) is an ordinal and α ∈ H(G(y))].
Thus H(G(y)) is an ordinal in P greater than each α < κ, so since P is transitive,
κ ∈ P.
An infinite cardinal κ is first-order describable iff there is a U ⊆ V κ and a sentence σ in
the language for (V κ , ∈, U) such that (V κ , ∈, U) |= σ, while there is no α < κ such that
(V α , ∈, U ∩ V α ) |= σ.
Theorem 18.14. If κ is infinite but not inaccessible, then it is first-order describable.
Proof. ω is describable by the sentence that says that κ is the first limit ordinal;
absoluteness is used. The subset U is not needed for this. Now suppose that κ is singular.
Let λ = cf(κ), and let f be a function whose domain is some ordinal γ < κ with
rng(f) cofinal in κ. Let U = {(λ, β, f(β)) : β < λ}. Let σ be the sentence expressing the
following:
For every ordinal γ there is an ordinal δ with γ < δ, U is nonempty, and there is an
ordinal µ and a function g with domain µ such that U consists of all triples (µ, β, g(β))
with β < µ.
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