ωι-Sonslin trees under countable support iterations
نویسندگان
چکیده
منابع مشابه
Many countable support iterations of proper forcings preserve Souslin trees
We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support iterations in the so-called Case A that does not need a division into forcings that add reals and those who do not.
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A major theme of [12] is preservation theorems for iterated forcing. These are theorems of the form “if 〈Pξ : ξ ≤ κ〉 is a countable support forcing iteration based on 〈Q̇ξ : ξ < κ〉 and each Q̇ξ has property such-and-such then Pκ has property thus-and-so.” The archetypal preservation theorem is the Fundamental Theorem of Proper Forcing [12, chapter III], which states that if each Q̇ξ is proper in V...
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We prove that any countable support iteration formed with posets with ω2-p.i.c. has ω2-c.c., assuming CH in the ground model. This improves earlier results of Shelah by removing the restriction on the length of the iteration. Thus, we solve the problem of obtaining a large continuum via such forcing iterations.
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In [4] a preservation theorem for countable support iterated forcing is proved with restriction to forcing notions which are not ω-distributive. We give the proof of the theorem without this restriction. 1. The preservation theorem. In [4] a preservation theorem for countable support iteration of proper forcing notions was proved with the additional assumption that all forcing notions which are...
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If ≤ is a preorder on a set P and p0 ≤ p1, we say that p1 is an extension of p0. Recall that a preorder is separative if and only if whenever p1 is not an extension of p0 there is an extension of p1 which is incompatible with p0. We say that P = (P,≤) is a forcing notion (also forcing poset) if ≤ is a separative preorder with minimal element 0P. Note that if P is separative and p1 p̌0 ∈ Ġ then p...
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ژورنال
عنوان ژورنال: Fundamenta Mathematicae
سال: 1993
ISSN: 0016-2736,1730-6329
DOI: 10.4064/fm-142-3-257-261