Abstract
We use the forcing with overlapping extenders (Gitik in Blowing up the power of a singular cardinal of uncountable cofinality, to appear in JSL) to give a direct construction of a model of \(\lnot \)SCH+Reflection.
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Notes
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 \(\bar{\kappa }_\eta ^+\) is preserved in \(V^{\langle \mathcal P,\le \rangle }\).
This is the crucial difference from the long extenders Prikry forcing \(\langle \mathcal P,\le , \le ^*\rangle \) of Section 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 this allowed Asaf Sharon [9] to build a non-reflecting stationary set. In the present setting both parts are put into one of cardinality \(\kappa _n\).
References
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Ben-Neria, O., Hayut, Y., Unger, S.: Stationary reflection and the failure of SCH
Gitik, M.: Prikry type forcings. In: Foreman, K. (ed.) Handbook of Set Theory, vol. 2, pp. 1351–1448. Springer, Berlin (2010)
Gitik, M.: Blowing up the power of a singular cardinal of uncountable cofinality (to appear in JSL)
Gitik, M.: An other method for constructing models of not approachability and not SCH
Gitik, M.: An other model with tree property and not SCH
Hayut, Y., Unger, S.: Stationary reflection
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Sharon, A.: Ph.D. thesis, Tel Aviv University (2005)
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The work was partially supported by Israel Science Foundation Grant No. 1216/18. We are grateful to the referee of the paper for his helpful remarks.
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Gitik, M. Reflection and not SCH with overlapping extenders. Arch. Math. Logic 61, 591–597 (2022). https://doi.org/10.1007/s00153-021-00805-3
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DOI: https://doi.org/10.1007/s00153-021-00805-3