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The tree property at $\aleph _{\omega +2}$ with a finite gap

Volume 251 / 2020

Sy-David Friedman, Radek Honzik, Šárka Stejskalová Fundamenta Mathematicae 251 (2020), 219-244 MSC: Primary 03E35; Secondary 03E55. DOI: 10.4064/fm866-2-2020 Published online: 19 June 2020


Let $n$ be a natural number, $2 \le n \lt \omega $. We show that it is consistent to have a model of set theory where $\aleph _\omega $ is strong limit, $2^{\aleph _\omega } = \aleph _{\omega +n}$, and the tree property holds at $\aleph _{\omega +2}$; we use a hypermeasurable cardinal of an appropriate degree and a variant of the Mitchell forcing followed by the Prikry forcing with collapses.


  • Sy-David FriedmanKurt Gödel Research Center for Mathematical Logic
    Währinger Straße 25
    1090 Vienna, Austria
  • Radek HonzikDepartment of Logic
    Charles University
    Celetná 20
    Praha 1, 116 42, Czech Republic
  • Šárka StejskalováInstitute of Mathematics AV ČR
    Žitná 25
    Praha 1, 115 67, Czech Republic

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