On changing cofinality of partially ordered sets

It is a subset of P V (A n ) consisting of at most ℵ k ∗ elements. But |A n | ≤ ℵ n .
Let us argue that there is Y ⊆ P V (A n ), Y ∈ V, |Y | ≤ ℵ k ∗
such that Y ⊇ {A s n | s ∈ S}.
Thus, |P V (A n )| V < ℵ ω , since ℵ ω is a strong limit cardinal in V . Let |P V (A n )| V = ℵ t , for
some t < ω. Fix
F : P V (A n ) ←→ ℵ t , F ∈ V.
Set Z 0 = F ′′ Z. Clearly, it’s enough to find some Z ∗ ∈ V, |Z ∗ | V ≤ ℵ k ∗ such that Z ∗ ⊇ Z 0 .
If t ≤ k ∗ , then just take Z ∗ = ℵ t . Suppose that t > k ∗ . Then there is η 0 < ℵ t such
that Z 0 ⊆ η 0 , since ℵ t is a regular cardinal in W . Find t 1 < t such that |η 0 | = ℵ t1 . Let
F 1 : η 0 ←→ ℵ t1 be a witnessing function in V . Set Z 1 = F 1 ′′ Z 0 . Clearly, it’s enough to find
some Z ∗ ∈ V, |Z ∗ | V ≤ ℵ k ∗ such that Z ∗ ⊇ Z 1 , since then F1
−1′′
Z ∗ ∈ V will be a desired cover
of Z 0 . If t 1 ≤ k ∗ , then just take Z ∗ = ℵ t1 . If t 1 > k ∗ , then there is η 1 < ℵ t1 such that
Z 1 ⊆ η 1 , since ℵ t1
is a regular cardinal in W . Continue the process. After finitely many
steps we must drop below k ∗ which in turn will provide a cover.
By shrink Y in V further if necessary we can assume that for every X ∈ Y there is t ∈ P
such that x ≼ t, for every x ∈ X. Working in V , we pick for each X ∈ Y an element a X ∈ P
such that x ≼ a X , for every x ∈ X.
Consider now the set
{a X | X ∈ Y }.
Clearly, it is in V , is cofinal for A n and has cardinality at most ℵ k ∗. So we are done.
□ of the claim.
Now, work in V and for each n < ω pick E n
cardinality at most ℵ k ∗. Then
E = ⋃ n