Which Pairs of Cardinals Can Be Hartogs and Lindenbaum Numbers of a Set?

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.