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Perfect subtree property for weakly compact cardinals

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Abstract

We investigate the consistency strength of the statement: κ is weakly compact and there is no tree on κ with exactly κ+ many branches. We show that this statement fails strongly (in the sense that there is a sealed tree with exactly κ+ many branches) if there is no inner model with a Woodin cardinal. Moreover, we show that for a weakly compact cardinal κ the nonexistence of a tree on κ with exactly κ+ many branches and, in particular, the Perfect Subtree Property for κ, implies the consistency of ADℝ + DC.

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Author information

Authors and Affiliations

  1. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel

    Yair Hayut

  2. Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstraße 8-10/104, 1040, Wien, Austria

    Sandra Müller

Authors
  1. Yair Hayut
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  2. Sandra Müller
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Corresponding author

Correspondence to Sandra Müller.

Additional information

The first-listed author was supported by Austrian Science Fund (FWF) Lise Meitner grant 2650-N35.

The second-listed author was supported by L’ORÉAL Austria, in collaboration with the Austrian UNESCO Commission and in cooperation with the Austrian Academy of Sciences — Fellowship Determinacy and Large Cardinals and Elise Richter grant number V844 of the FWF.

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Hayut, Y., Müller, S. Perfect subtree property for weakly compact cardinals. Isr. J. Math. 253, 865–886 (2023). https://doi.org/10.1007/s11856-022-2385-4

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  • DOI: https://doi.org/10.1007/s11856-022-2385-4

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