On changing cofinality of partially ordered sets

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then • for each 1 ≤ k < ω, (⃗r ∗ (κ 1 ) ⌢ ν(k)) 0 = r(k, ν) 0 ; • for each k, m < ω, 1 ≤ k, m let (⃗r ∗ (κ 1 ) ⌢ ν(k)) m = (r ∗ (k, κ 1 )) m . Now we have the following: for each ν < κ 1 , if there is a good sequence 〈r ′ (n) | n < ω〉 such that • 〈r ′ (n) | n < ω〉 ≥ ⃗r ∗ (κ 1 ) ⌢ ν, and • r ′ (1) 〈ν〉 ‖ Q (〈ν, T ∼ (〈r ′ (2) 〈ν〉 , ..., r ′ (n) 〈ν〉 , ... | 2 ≤ n < ω〉)〉‖σ), ⃗r ∗ (κ 1 ) ⌢ ν(1)) 〈ν〉 ‖ Q (〈ν, T ∼ (〈⃗r ∗ (κ 1 ) ⌢ ν(2) 〈ν〉 , ..., ⃗r ∗ (κ 1 ) ⌢ ν(n) 〈ν〉 , ... | 2 ≤ n < ω〉)〉‖σ). This defines a splitting of κ 1 into three sets. By shrinking to a set in U 1 , if necessary, we assume that every ν < κ 1 is in the same part of the partition. Suppose that we are in the situation in which σ is forced by every ν < κ 1 . Define then a good sequence ⃗q = 〈q(n) | n < ω〉 which is above ⃗p and forces σ, i.e. Thus, q(0)‖ Q (〈〈〉, T (⃗q)〉‖ σ). • for each m, 1 ≤ m < ω set q(1) m = j 1 ((r ∗ (1, κ 1 )) m ), • (q(0)) 0 = π ≤0,≤1 ′′ [〈r(1, ν) 0 | ν < κ 1 〉] U1 ; • for each 1 ≤ k < ω, (q(k)) 0 = π ≤k,≤k+1 ′′ [〈r(k + 1, ν) 0 | ν < κ 1 〉] U1 ; • for each k, m < ω, 1 ≤ k, m let q(k + 1) m = j 1 ((r ∗ (k, κ 1 )) m ). If σ is not decided a set of ν’s in U 1 , then we proceed similar, but deal with pairs 〈ν 1 , ν 2 〉 ∈ κ 1 × κ 2 , triples 〈ν 1 , ν 2 , ν 3 〉 ∈ κ 1 × κ 2 × κ 3 etc., instead of ν ∈ κ 1 . At certain level a decision about σ will be made and then we will be able to go back down and the contradiction will be derived. □ Our final goal will be to show that the forcing with P over V [G] preserves cofinalities. The usual Prikry argument does not work here directly due to the lack of closure (beyond ℵ 1 ). The idea will be to redo the proof of the Prikry property while splitting Q into Q >n and Q ≤n , such that Q >n has enough closure and Q ≤n satisfies enough chain condition. Let us start by showing that all cardinals below κ 1 (and then also κ 1 are preserved. 26
Lemma 4.24 〈P, ≤ 〉 preserves all the cardinals below κ 1 . Proof. Work in V with Q ∗ P . Let µ < λ < κ 1 be cardinals, λ a regular cardinal and let h ∼ be a Q ∗ P name of a function from µ to λ, as forced by the weakest condition. Let ⃗p be a good sequence. As in Lemma 4.23 we find a good sequence ⃗q 00 ≥ ⃗p and δ 00 < λ such that q(0) 00 ‖ Q (〈〈〉, T (⃗q 00 〉‖ h ∼ (0) = δ 00 ). Note that λ < κ 1 and so the number of possible values for h ∼ is bounded in κ 1 . Hence on the set in U 1 we will have the same value. One can try now to do the same with h ∼ (1). But as a result ⃗q 00 may increase. If we continue further and go through all h ∼ (n), then due to the luck of closure of Q ≤0 (recall that conditions there are just finite) there may be no single condition stronger than all the constructed in the process. Let us instead continue to deal with h ∼ (0) and find ⃗q 0 ≥ ⃗p and δ 0 such that • q(0) 0 ‖ Q (〈〈〉, T (⃗q 0 〉‖ h ∼ (0) ≤ δ 0 ). • q(k) 0 0 = p(k) 0 , for each k < ω It is not hard to do just using (5) of Definition 4.12 and c.c.c. of Q ≤0 and its images. Just run the argument of 4.23 enough (< ω 1 ) times. Now, with ⃗q 0 and δ 0 we continue to h ∼ (1). Define similar ⃗q 1 and δ 1 so that • ⃗q 1 ≥ ⃗q 0 , • q(0) 1 ‖ Q (〈〈〉, T (⃗q 1 〉‖ h ∼ (1) ≤ δ 1 ), • q(k) 1 0 = p(k) 0 , for each k < ω Continue and define for each m < ω, ⃗q m and δ m so that • ⃗q m ≥ ⃗q m−1 , • q(0) m ‖ Q (〈〈〉, T (⃗q m 〉‖ h ∼ (m) ≤ δ m ), • q(k) m 0 = p(k) 0 , for each k < ω Finally, we can now put all this ⃗q m ’s together. There is a good sequence ⃗q such that • ⃗q ≥ ⃗q m , for every m < ω, 27
Page 1 and 2: On changing cofinality of partially
Page 3 and 4: as K V -the one computed in V . Jus
Page 5 and 6: It is a subset of P V (A n ) consis
Page 7 and 8: Consider (again in V ) the followin
Page 9 and 10: Force now a new ω sequence to each
Page 11 and 12: Now, for each α ∈ [κ, κ + ), w
Page 13 and 14: 3. the sequence 〈ν 0 , ν 1 , ..
Page 15 and 16: 2. n < m implies a n (q) ⊆ a m (q
Page 17 and 18: I.e. it is the set of all elements
Page 19 and 20: 4. β ∈ j ≤l,≤n (a 0 (q)) \ a
Page 21 and 22: 1. for every n, 0 < n < ω, [f p(n)
Page 23 and 24: Definition 4.18 Let 〈η, T (⃗p)
Page 25: Proof. Let σ be a statement and

On changing cofinality of partially ordered sets

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