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

Abstract

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

9 pages; accepted version

Cited by in corpus (1)

• The Hartogs-Lindenbaum Spectrum of Symmetric Extensions