Supercompactness and measurable limits of strong cardinals (Q2747707)

A Menas-type result is an instance of the following theorem: NEWLINENEWLINENEWLINE(MTR) ``Let \({\mathbf P}\) be a definable property of ordinals, \(\alpha < \kappa\) and let \(\kappa\) be the \(\alpha\)th measurable limit of cardinals with property \({\mathbf P}\). Then \(\kappa\) is not \(2^\kappa\)-supercompact.'' NEWLINENEWLINENEWLINEMenas-type results go back to work of \textit{T. K. Menas} [Ann. Math. Logic 7, 327-359 (1975; Zbl 0299.02084)] and belong to the investigation of the difference between the notions of supercompactness and strong compactness: Using (MTR) for \({\mathbf P} =\) ``is strongly compact'', Menas proved that the \(\alpha\)th measurable limit of strongly compact cardinals is strongly compact but cannot be supercompact. NEWLINENEWLINENEWLINEThe author investigated the sharpness of Menas' result in his paper [J. Symb. Log. 64, 1675-1688 (1999; Zbl 0959.03041)] and in joint work with \textit{S. Shelah} [Trans. Am. Math. Soc. 349, 2007-2034 (1997; Zbl 0876.03030)] for \({\mathbf P} =\) ``is strongly compact'' and \({\mathbf P} =\) ``is supercompact''. NEWLINENEWLINENEWLINEIn the paper under review, Apter looks at Menas-type results with \({\mathbf P} = \) ``is strong'' and is able to get results corresponding to the mentioned results in the \({\mathbf P} = \) ``is supercompact'' case. NEWLINENEWLINENEWLINEIn the words of the author: ``Since ideas and notions from [the mentioned papers by Apter and Apter-Shelah] are used throughout the course of this paper, it would be most helpful to readers if copies of these papers were kept close at hand (p. 631)''. A thorough knowledge of these papers is necessary to understand motivation and proofs of the paper under review. NEWLINENEWLINENEWLINEThe results of this paper have been improved by the author in a more recent paper [J. Symb. Log. 66, 1919-1927 (2001; Zbl 0994.03045)].