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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Collapsing successors of singulars

Author: James Cummings
Journal: Proc. Amer. Math. Soc. 125 (1997), 2703-2709
MSC (1991): Primary 03E05; Secondary 03E35
MathSciNet review: 1416080
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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. 1) $W$ is not a $\kappa ^+$-c.c generic extension of $V$. 2) There is no ``good scale for $\kappa $'' in $V$, so in particular weak forms of square must fail at $\kappa $. 3) If $V \models \operatorname {cf}(\kappa ) = \aleph _0$ then $V \models \hbox {``$\kappa $ is strong limit $\implies 2^\kappa = \kappa ^+$'',}$ and also ${}^\omega \kappa \cap W \supsetneq {}^\omega \kappa \cap V$. 4) If $\kappa = \aleph _\omega ^V$ then $\lambda \le (2^{\aleph _0})_V$.

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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

Keywords: Squares, pcf, successors of singulars, good points
Received by editor(s): March 20, 1996
Communicated by: Andreas R. Blass
Article copyright: © Copyright 1997 American Mathematical Society