On a problem of Woodin
type: Item
instance of: scholarly article
title: On a problem of Woodin
zbMATH Open document ID: 0958.03034
published in: Archive for Mathematical Logic
publication date: 2001-03-22T00:00:00Z
review text: In \textit{A. Kanamori}'s book [The higher infinite, Springer-Verlag, Berlin (1994; Zbl 0813.03034)], the following question of Woodin's is mentioned: If \(\kappa\) is a strongly compact cardinal and \(2^\delta= \delta^+\) for every cardinal \(\delta< \kappa\), does this entail that GCH holds? In this paper the author shows that in the absence of full AC it is possible to get a negative answer to Woodin's question. Specifically he proves: Let \(V\models \text{``ZFC}+ \kappa\) is supercompact''. Then there is a partial ordering \(P\in V\) and a symmetric inner model \(N\), \(V\subseteq N\subseteq V^P\), so that \(N\models \text{``ZF}+ \forall\delta< \kappa\) \([DC_\delta]+ \kappa\) is a strong limit cardinal \(+\) \(\forall\delta< \kappa\) \([2^\delta= \delta^+]+ \kappa\) is supercompact + There is a sequence \(\langle A_\alpha: \alpha< \kappa^{++}\rangle\) of distinct subsets of \(\kappa\)''.
reviewed by: Egbert Harzheim
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 03E25
zbMATH DE Number: 1471392
zbMATH Keywords: strongly compact cardinal
zbMATH Keywords: GCH
zbMATH Keywords: AC
author: Arthur W. Apter
MaRDI profile type: Publication
full work available at URL: s001530050146
OpenAlex ID: W2095302653
DOI: 10.1007/S001530050146
description: scientific article; zbMATH DE number 1471392