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Some weak fragments of Martin’s axiom related to the rectangle refining property. (English) Zbl 1153.03038

A forcing notion \(\mathbb P\) is said to have the anti-rectangle refining property if it is uncountable, and given \(I,J \in [\mathbb P]^{\aleph_1}\), there are \(I'\in [I]^{\aleph_1}\) and \(J'\in [J]^{\aleph_1}\) such that \(p\) and \(q\) are incompatible in \(\mathbb P\) for all \(p\in I'\) and \(q\in J'\). It is shown that, as a forcing notion, an Aronszajn tree has the anti-rectangle refining property, and so has a forcing notion freezing an \((\omega_1, \omega_1)\)-gap. Moreover, it is shown to be consistent that MA\(_{\aleph_{1}}({\mathcal P})\) holds but there exists and entangled set of reals (and hence MA\(_{\aleph_{1}}\) fails), where \({\mathcal P} = \{ a(\mathbb P) : \mathbb P\) has the anti-rectangle refining property\(\}\) (\(a (\mathbb P)\) is the set of all finite antichains in \(\mathbb P\), ordered by reverse inclusion).

MSC:

03E50 Continuum hypothesis and Martin’s axiom
03E35 Consistency and independence results
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References:

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