On changing cofinality of partially ordered sets

Consider (again in V ) the following set:
Y ∗ = {B ∈ Y | ∃ν∀τ ∈ B
τ ≼ ν}.
Then Y ∗ still includes X, since each element of X has such property. Remember that S is
cofinal. In particular, for each τ ∈ A there is i < η such that τ ≼ η i and, hence τ ∈ A i .
Then, in V , the following holds:
(*) for each τ ∈ A there is B ∈ Y ∗ with τ ∈ B.
Now working in V we pick for each B ∈ Y ∗ some ν(B) such that τ ≼ ν(B), for each
τ ∈ B. Set
T = {ν(B) | B ∈ Y ∗ }.
Then T ∈ V , |T | ≤ |Y ∗ | < |A| and by (*) we have that for each τ ∈ A there is ν ∈ T with
τ ≼ ν. Hence, T witnesses cof P (A) < |A|.
□ of the lemma.
Work in V . Let us prove that for each A ⊆ κ (cof P (A)) V ≤ η, by induction on |A|.
If |A| ≤ η then this is trivial. Suppose that the statement is true for each cardinal less than
ρ. Let us prove it for ρ. Let A ⊆ κ of cardinality ρ. If ρ = µ + , for some µ, then by Lemma
2.8 we have cof P (A) < |A| = ρ. Suppose that ρ is a limit cardinal. If cof(ρ) > η, then again
by Lemma 2.8 we have cof P (A) < |A| = ρ. Let finally ρ be a limit cardinal of cofinality
at most η. Pick a cofinal in ρ sequence 〈ρ i | i < η〉. We present A as a union of sets A i ,
i < η such that |A i | = ρ i . Apply the induction to each of A i ’s. We find T i of cardinality η
witnessing cof P (A i ) ≤ η. Set T = ⋃ i