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Archive for Mathematical Logic

, Volume 58, Issue 1–2, pp 1–26 | Cite as

Diagonal reflections on squares

  • Gunter FuchsEmail author
Article
  • 31 Downloads

Abstract

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 \(\lambda \ge \omega _2\), 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.

Keywords

Square principles Stationary reflection Forcing axioms Subcomplete forcing Resurrection axioms Continuum hypothesis 

Mathematics Subject Classification

03E50 03E57 03E35 03E55 03E05 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The College of Staten Island (CUNY)Staten IslandUSA
  2. 2.The Graduate Center (CUNY)New YorkUSA

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