Abstract
Let \(D\) be an infinite discrete set of measurable cardinals. It is shown that generalized Prikry forcing to add a countable sequence to each cardinal in \(D\) is subcomplete. To do this it is shown that a simplified version of generalized Prikry forcing which adds a point below each cardinal in \(D\), called generalized diagonal Prikry forcing, is subcomplete. Moreover, the generalized diagonal Prikry forcing associated to \(D\) is subcomplete above \(\mu \), where \(\mu \) is any regular cardinal below the first limit point of \(D\).
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Notes
Collection is the following schema: \(\forall \mathbf {w}\, [\forall x \, \exists y \; \varphi (x, y, \mathbf {w}) \implies \forall A \exists B \, \forall x \in A \,\exists y \in B \; \varphi (x, y, \mathbf {w}) ]\).
See [4, Section 3.1 Lemma 2.4].
As Jensen [4, Lemma 5.3] explains, this is exactly why the third requirement of subcompleteness involves this type of Skolem hull - these liftups can be recovered.
Recall that \(\delta _{N_*}\) is the least ordinal for which \(L_{\delta _{N_*}}(N_*)\) is admissible.
As Fuchs [1, p. 966] points out, this result is a modification to the proof that generalized diagonal Prikry forcing preserves cardinalities.
References
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Fuchs, G.: The subcompleteness of Magidor forcing. Arch. Math. Logic 57, 273–284 (2017)
Jensen, R.B.: Subproper and subcomplete forcing. Handwritten notes (2009). https://www.mathematik.hu-berlin.de/~raesch/org/jensen.html
Jensen, R.B.: Subcomplete Forcing and \({\cal{L}}\)-Forcing. In: Chitat, C., Feng, Q., Slaman, T.A., Woodin, W.H., Yang, Y. (eds.) E-recursion, forcing and \(C^*\)-algebras, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, pp. 83–182. World Scientific (2014)
Minden, K.: On Subcomplete forcing. Ph.D. thesis, The Graduate Center of the City University of New York (2017). https://arxiv.org/abs/1705.00386
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The material presented here is based on the author’s doctoral thesis [5], written at the Graduate Center of CUNY under the supervision of Gunter Fuchs.
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Minden, K. The subcompleteness of diagonal Prikry forcing. Arch. Math. Logic 59, 81–102 (2020). https://doi.org/10.1007/s00153-019-00678-7
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DOI: https://doi.org/10.1007/s00153-019-00678-7