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

Given any $\lambda \le \kappa$, we construct a symmetric extension in which there is a set $X$ such that $\mathrm{\aleph }(X)=\lambda$ and ${\mathrm{\aleph }}^{\ast }(X)=\kappa$. Consequently, we show that $\mathsf{Z}\mathsf{F}+$“For all pairs of infinite cardinals $\lambda \le \kappa$ there is a set $X$ such that $\mathrm{\aleph }(X)=\lambda \le \kappa ={\mathrm{\aleph }}^{\ast }(X)$” is consistent.