Extender based forcings, fresh sets and Aronszajn trees

نویسنده

  • Moti Gitik
چکیده

Extender based forcings are studied with respect of adding branches to Aronszajn trees. We construct a model with no Aronszajn tree over אω+2 from the optimal assumptions. This answers a question of Friedman and Halilović [1]. The reader interested only in Friedman and Halilović question may skip the first section and go directly to the second. 1 No branches to κAronszajn trees. We deal here with Extender Based Prikry forcing, Long and short extenders Prikry forcing. Let us refer to [2] for definitions. Theorem 1.1 Extender based Prikry forcing over κ cannot add a cofinal branch to a κAronszajn tree. Proof. Let 〈T,≤T 〉 be a κ-Aronszajn tree. Denote by P the extender based Prikry forcing over κ. Suppose that P adds a cofinal branch through T . Let b ∼ be a name of such branch and 0P b ∼ is a κ -branch through T . Let p, q ∈ P and n < ω. We say that qis an n-extension of p iff q ≥ p and q is obtained from p by taking n-element sequence 〈η1, ..., ηn〉 from the first n-levels of the tree of sets of measures one over the maximal coordinate of p, adding it to p and projecting to all permitted coordinates of p. Denote such q by p〈η1, ..., ηn〉. For each α < κ and p ∈ P there are n < ω and pα ≥∗ p such that any n-extension of pα decides b ∼(α), as was shown in [5]. Note that the branch b ∼ α + 1 is decided as well, since T ∈ V and so the value at the level α determines uniquely the branch to it below. Denote by n(p, α) the least such n.

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تاریخ انتشار 2011