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Sigma-Prikry forcing I: The Axioms
Published online by Cambridge University Press: 26 May 2020
Abstract
We introduce a class of notions of forcing which we call $\Sigma $ -Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $\Sigma $ -Prikry. We show that given a $\Sigma $ -Prikry poset $\mathbb P$ and a name for a non-reflecting stationary set T, there exists a corresponding $\Sigma $ -Prikry poset that projects to $\mathbb P$ and kills the stationarity of T. Then, in a sequel to this paper, we develop an iteration scheme for $\Sigma $ -Prikry posets. Putting the two works together, we obtain a proof of the following.
Theorem. If $\kappa $ is the limit of a countable increasing sequence of supercompact cardinals, then there exists a forcing extension in which $\kappa $ remains a strong limit cardinal, every finite collection of stationary subsets of $\kappa ^+$ reflects simultaneously, and $2^\kappa =\kappa ^{++}$ .
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- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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- © Canadian Mathematical Society 2020
Footnotes
Poveda was partially supported by the Spanish Government under grant MTM2017-86777-P, by Generalitat de Catalunya (Catalan Government) under grant SGR 270-2017 and by MECD Grant FPU15/00026. Rinot was partially supported by the European Research Council (grant agreement ERC-2018-StG 802756) and by the Israel Science Foundation (grant agreement 2066/18). Sinapova was partially supported by the National Science Foundation, Career-1454945.
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