Which pairs of cardinals can be Hartogs and Lindenbaum numbers of a set?
Volume 267 / 2024
Fundamenta Mathematicae 267 (2024), 231-241
MSC: Primary 03E25; Secondary 03E35
DOI: 10.4064/fm231006-14-8
Published online: 7 November 2024
Abstract
Given any $\lambda \leq \kappa $, we construct a symmetric extension in which there is a set $X$ such that $\aleph (X)=\lambda $ and $\aleph ^*(X)=\kappa $. Consequently, we show that $\mathsf{ZF} {}+{}$“for all pairs of infinite cardinals $\lambda \leq \kappa $ there is a set $X$ such that $\aleph (X)=\lambda \leq \kappa =\aleph ^*(X)$” is consistent.
