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Diagonal reflections on squares. (English) Zbl 07006123

Summary: The effects of (bounded versions of) the forcing axioms \(\mathsf {SCFA}\), \(\mathsf {PFA}\) and \(\mathsf {MM}\) on the failure of weak threaded square principles of the form \(\square (\lambda ,\kappa )\) are analyzed. To this end, a diagonal reflection principle, \(\mathsf {DSR}{\left( {<}\kappa ,S\right) }\) is introduced. It is shown that \(\mathsf {SCFA} \) implies \(\mathsf {DSR}{\left( \omega _1,S^\lambda _\omega \right) }\), for all regular \(\lambda \geq \omega _2\), and that \(\mathsf {DSR}{\left( \omega _1,S^\lambda _\omega \right) }\) implies the failure of \(\square (\lambda ,\omega _1)\) if \(\lambda >\omega _2\), and it implies the failure of \(\square (\lambda ,\omega )\) if \(\lambda =\omega _2\). It is also shown that this result is sharp. It is noted that \(\mathsf {MM}\)/\(\mathsf {PFA}\) imply the failure of \(\square (\lambda ,\omega _1)\), for every regular \(\lambda >\omega _1\), and that this result is sharp as well.

MSC:

03E50 Continuum hypothesis and Martin’s axiom
03E57 Generic absoluteness and forcing axioms
03E35 Consistency and independence results
03E55 Large cardinals
03E05 Other combinatorial set theory
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