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

TL;DR

This paper constructs models in set theory without choice where sets can have prescribed Hartogs and Lindenbaum numbers, demonstrating the consistency of certain cardinal configurations.

Contribution

It introduces a method to realize any pair of infinite cardinals as the Hartogs and Lindenbaum numbers of a set within a symmetric extension.

Findings

Existence of sets with prescribed Hartogs and Lindenbaum numbers.

Consistency of the statement that all pairs of infinite cardinals can be realized as such numbers.

Construction of symmetric extensions for arbitrary infinite cardinal pairs.

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.