Abstract
We prove a Mathias-type criterion for the Magidor iteration of Prikry forcings.
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Notes
i.e., obtained by choosing the next n-points of the generic Prikry sequence.
\({{\,\textrm{supp}\,}}(p^*)\) is the support of \(p^*\) from Definition 1.5 (3).
Note that \(\kappa ^{T}(s)\) can be determined from T and s since \(\kappa ^{T}(s) = \sup ({{\,\textrm{Succ}\,}}_T(s))\).
Such conditions exist by Fact 1.7.
The map \(k^F\) associated with the rest of the iteration \(\mathcal {T}\) over F, does not overlap \(\kappa _0^F\) and consists of critical points below \(\kappa _0^F\) or strictly above \(\kappa _0^F\). Since \(\kappa _0^F \in \Delta \) is inaccessible, \(k^F(\kappa _0^F) = \kappa _0^F\).
\(C^{M^X_i}_{\bar{\kappa }}\) could be finite.
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Acknowledgements
The author would like to thank Philip Welch for raising the question concerning the Magidor iterations and genericity of sequences of generator in associated iterated ultrapowers, and to Philip Welch and Christopher Turner for pointing out a gap in a related argument of the author from [4] (Lemma 4.5). He is grateful to Moti Gitik for his remark concerning Example 1.9, to Christopher Turner for pointing out several problems in an earlier version of this paper, and to the referee for suggestions and corrections that improved the clarity of the paper.
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Ben-Neria, O. A Mathias criterion for the Magidor iteration of Prikry forcings. Arch. Math. Logic 63, 119–134 (2024). https://doi.org/10.1007/s00153-023-00887-1
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DOI: https://doi.org/10.1007/s00153-023-00887-1