1 Ju l 1 99 2 REMARK ON THE FAILURE OF MARTIN ’ S AXIOM
نویسنده
چکیده
Let m be the least cardinal θ such that MA θ fails. The only known model for " m is singular " was constructed by Kunen [K1]. In Kunen's model cof (m) = ω 1. It is unknown whether " ω 1 < cof (m) < m " is consistent. The purpose of this paper is to present a proof of Kunen's result and to identify the difficulties of generalizing this result to an arbitrary uncountable cofinality. The following material is based on [K1]. We would like to thank K. Kunen and S. Shelah [S] for their input. §0 Definitions and some facts. For undefined terminology consult [K]. Let θ be a fixed singular cardinal with uncountable cofinality. Let κ = cof (θ), and fix {θ α : α < κ} an increasing sequence of cardinals converging to θ with θ 0 > κ. and (∀b ∈ Y)|c ∩ b| < ℵ 0 ]). Notice that if (A, B) is locally split in V, then (A, B) is locally split in every c.c.c. extension of V. For a disjoint pair (A, B) we define the partial order S(A, B) = {f : θ → ω : |f | < ℵ 0 and (∀ξ, η ∈ dom(f)) (a ξ −f (ξ))∩(b η −f (η)) = ∅}. can be split (to split (A, B) use the set c = {a ξ − f (ξ) : f ∈ G, ξ ∈ dom(f)} Where G is a sufficiently generic filter on S(A, B)).
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تاریخ انتشار 1992