Summary
The notion of an entangled linear order gives a useful method to construct counter examples for such problems as the productivity of chain conditions, the square bracket partition relation and the existence of a large size monotonic subfunction. In particular, if there exists and ℵ1-entangled linear order then some consequences ofMA ℵ 1 or of wOCA fail. So, in which model ofZFC does an ℵ1-entangled linear order exist? Todorcevic [6] has shown if cf2ℵ 0=ω1 then there is an ℵ1-entangled linear order of size ℵ1. And it is widely known that adding κ many Cohen reals adds such a linear order of size κ. In this note it is shown that adding asingle Cohen real is sufficient to get a maximal size ℵ1-entangled linear order.
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Yuasa, Y. Adding a Cohen real adds an entangled linear order. Arch Math Logic 32, 299–304 (1993). https://doi.org/10.1007/BF01387408
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DOI: https://doi.org/10.1007/BF01387408