The tree property at the double successor of a singular cardinal with a larger gap
For a regular cardinal \(\kappa,\) the tree property at \(\kappa\) is the assertion that there are no \(\kappa\)-Aronszajn trees. The tree property belongs to a family of combinatorial assertions known as compactness principles and has received special attention in the last few years. In the paper under review, the authors study the tree property at the double successor of a singular strong limit cardinal. The first result in this direction is due to \textit{J. Cummings} and \textit{M. Foreman} [Adv. Math. 133, No. 1, 1--32 (1998; Zbl 0949.03039)], who produced, starting from a supercompact cardinal \(\kappa\) and a weakly compact cardinal above it, a model of ZFC in which \(\kappa\) is a singular strong limit cardinal of countable cofinality and the tree property holds at \(\kappa^{++}.\) They also extended their result for \(\kappa=\aleph_{\omega}\). \textit{S.-D. Friedman} and \textit{A. Halilović} [J. Symb. Log. 76, No. 2, 477--490 (2011; Zbl 1221.03053)] have obtained the same results from the weaker assumption of a cardinal \(\kappa\) which is \(H(\lambda)\)-hypermeasurable for some weakly compact cardinal \(\kappa.\) In all these models, \(2^\kappa = \kappa^{++}\) and it is natural to ask if we can obtain the tree property at the double successor of a singular strong limit cardinal \(\kappa\) with \(2^\kappa > \kappa^{++}\). In this paper, the authors prove such a result. Starting from a supercompact cardinal \(\kappa\) and a weakly compact cardinal \(\lambda\) above it, they build a generic extension in which \(\kappa\) is a singular strong limit cardinal of countable cofinality, \(\lambda=\kappa^{++}\), the tree property holds at \(\kappa^{++}\) and \(2^\kappa=\lambda^+=\kappa^{+++}.\) They also sketch how one can make \(2^\kappa\) arbitrary large. The method of the proof is in general based on the argument of Cummings and Foreman [loc.cit.], but is more involved in details since the authors have to add \(\mu\)-many subsets of \(\kappa\) for some \(\mu > \lambda\) and the normal measure they take for the Priky part of the forcing lives in the extension by \(\mathrm{Add}(\kappa, \mu)\), and this introduces a mismatch between the length of the first and second component of the defined forcing. This problem is solved by reflecting the normal measure more carefully and in several stages. It is worth noting that the result of the paper can be proved from weaker large cardinals, as is shown by latter works of the same authors. For example, a cardinal \(\kappa\) which is \(H(\lambda^+)\)-hypermeasurable and \(\lambda\) is weakly compact is sufficient to get a model in which \(\kappa\) is a singular strong limit cardinal of countable cofinality, the tree property holds at \(\kappa^{++}\) and \(2^\kappa=\lambda^+=\kappa^{+++}\).
scientific article; zbMATH DE number 6856634