Collapsing successors of singulars
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- by James Cummings
- Proc. Amer. Math. Soc. 125 (1997), 2703-2709
- DOI: https://doi.org/10.1090/S0002-9939-97-03995-6
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Abstract:
Let $\kappa$ be a singular cardinal in $V$, and let $W \supseteq V$ be a model such that $\kappa ^+_V = \lambda ^+_W$ for some $W$-cardinal $\lambda$ with $W \models \operatorname {cf}(\kappa ) \neq \operatorname {cf}(\lambda )$. We apply Shelah’s pcf theory to study this situation, and prove the following results.
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$W$ is not a $\kappa ^+$-c.c generic extension of $V$.
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There is no “good scale for $\kappa$” in $V$, so in particular weak forms of square must fail at $\kappa$.
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If $V \models \operatorname {cf}(\kappa ) = \aleph _0$ then $V \models {}$ “$\kappa$ is strong limit $\implies 2^\kappa = \kappa ^+$”, and also ${}^\omega \kappa \cap W \supsetneq {}^\omega \kappa \cap V$.
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If $\kappa = \aleph _\omega ^V$ then $\lambda \le (2^{\aleph _0})_V$.
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Bibliographic Information
- James Cummings
- Affiliation: Department of Mathematics 2-390, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- Address at time of publication: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213-3890
- MR Author ID: 289375
- ORCID: 0000-0002-7913-0427
- Email: cummings@math.mit.edu, jcumming@andrew.cmu.edu
- Received by editor(s): March 20, 1996
- Communicated by: Andreas R. Blass
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2703-2709
- MSC (1991): Primary 03E05; Secondary 03E35
- DOI: https://doi.org/10.1090/S0002-9939-97-03995-6
- MathSciNet review: 1416080