On changing cofinality of partially ordered sets

2. n < m implies a n (q) ⊆ a m (q).
3. Let α, β < κ and n, 1 ≤ n < ω be the least such that α < κ + n . Then
α ≺ q,0 β implies that
• if k, 1 ≤ k < ω is the least such that β < κ + k
, then k ≥ n,
• 〈α, β〉 ∉ ⊤ q,n .
Define the forcing order on Q as follows.
Definition 4.2 Let p, q ∈ Q. Set p ≥ Q q iff for each n < ω
1. a n (p) ⊇ a n (q)
2. ≼ p,0 ∩[a 0 (q)] 2 =≼ q,0
3. ⊤ p,n ∩ [a n (q)] 2 = ⊤ q,n
For each n < ω let Q >n consists of all 〈p m | ω > m > n〉 such that for some 〈p k | k ≤ n〉
we have 〈p i | i < ω〉 ∈ Q.
Let G >n be a generic subset of Q >n . Define Q ≤n to be the set of all sequences 〈p k | k ≤ n〉
such that for some 〈p m | ω > m > n〉 ∈ G >n we have 〈p i | i < ω〉 ∈ Q.
The next lemma is immediate.
Lemma 4.3 For each n < ω
1. the forcing Q >n is κ + n+1-closed,
2. the forcing Q ≤n satisfies κ ++
n -c.c. in V Q >n
,
3. Q ≃ Q >n ∗ Q ≤n .
Let G be a generic subset of Q. Work in V [G].
Set
Define P = 〈κ, ≼ 〉.
≼= ⋃ {≼ q,0 | q ∈ G}.
Lemma 4.4 Let A ∈ V be a subset of κ + n
cof P (A) = κ + n .
of cardinality κ + n , for some n, 0 < n < ω. Then
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