On changing cofinality of partially ordered sets

But Y ∈ U m implies κ m ∈ j m (Y ). Hence we have in M m an element of j m (Y ) which is above
j m (α). Then, by the elementarity, there is an element of Y which is above α. Contradiction.
□
Similar and unfortunately, the following holds as well.
Lemma 4.10 Let n < m < ω, α < κ + n . Then the set
is U m -positive.
{β < κ m | α ⊥ β}
Remark 4.11 Note that the proof of Lemma 4.9 provides a bit more information. Thus, if
r = j m (q) and α < κ + n does not belong to a q,0 , then r can be extended to a condition p by
either adding α ≺ p,0 κ m or α ⊥ p,0 κ m . Just Definition 4.1 puts no restrictions here.
Let us define now a forcing P similar to the diagonal Prikry forcing. Instead of sets of
measure one positive sets will be used. Also a small addition will be made in order to insure
that a countable cofinal subset will be added to P.
Definition 4.12 A sequence ⃗p = 〈p(0), p(1), ..., p(n), ...〉 will be called a good sequence iff
1. p(0) ∈ Q.
For every l, n, l < n < ω the following hold:
2. p(n) ∈ Q ∗∗