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Collapsing successors of singulars

Authors:	James Cummings

Institution:	Department of Mathematics 2-390, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Abstract:	Let $\kappa$ be a singular cardinal in , and let $W \supseteq V$ be a model such that $\kappa ^+_V = \lambda ^+_W$ for some -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) is not a $\kappa ^+$ -c.c generic extension of . 2) There is no ``good scale for $\kappa$ ' in , 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 $\lambda \le (2^{\aleph _0})_V$ .

Keywords:	Squares pcf successors of singulars good points

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