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Another method for constructing models of not approachability and not SCH

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Abstract

We present a new method of constructing a model of \(\lnot \)SCH+\(\lnot \)AP.

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Notes

  1. Section 3 of [5] contains an essential flow, which is due solely to the first author, but it turns out that with the forcing of [4], it is possible to make the idea work.

  2. This condition basically says that one entree given dense open set by taking a direct extension and then specifying finitely many coordinates. Usually, this property has the same proof, as the Prikry condition and is used to show that \(\lambda ^+\) is preserved in \(V^{{\langle }{{\mathcal {P}}},\le {\rangle }}\).

  3. This is the crucial difference from the long extenders Prikry forcing \({\langle }{{\mathcal {P}}}\le , \le ^*{\rangle }\) of Sec. 2 of [3] The conditions in \({{\mathcal {P}}}\) consist basically of two parts one of cardinality \(<\kappa _n, (n<\omega )\) (assignment functions) and another of cardinality \(\kappa _\omega \) (Cohen functions). As a result, \({\langle }{{\mathcal {P}}},\le ^*{\rangle }\) collapses \(\kappa _\omega ^+\) and, as Asaf Sharon pointed out, \({\langle }{{\mathcal {P}}},\le , \le ^*{\rangle }\) adds \(\square ^*_{\kappa _\omega }\).

    In the present setting both parts are put into one of cardinality \(\kappa _n\).

  4. Either the Magidor or Easton support can be used for this.

References

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  1. Tel Aviv University, Tel Aviv, Israel

    Moti Gitik

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  1. Moti Gitik
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Correspondence to Moti Gitik.

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The work was partially supported by Israel Science Foundation Grants 58/14, 1216/18. We are grateful to the referee of the paper for her/his remarks and corrections.

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Gitik, M. Another method for constructing models of not approachability and not SCH. Arch. Math. Logic 60, 469–475 (2021). https://doi.org/10.1007/s00153-020-00755-2

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  • DOI: https://doi.org/10.1007/s00153-020-00755-2

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