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arXiv · 2309.11409

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

Asaf Karagila·Calliope Ryan-Smith
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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.

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Asaf Karagila, Calliope Ryan-Smith. 2023-09-20. Which Pairs of Cardinals Can Be Hartogs and Lindenbaum Numbers of a Set?. https://doi.org/10.4064/fm231006-14-8

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