A generalisation of Läuchli's lemma

• Published in: Mathematical logic quarterly 2024-05, Vol.70 (2), p.173-177
• Main Authors: Sonpanow, Nattapon, Vejjajiva, Pimpen
• Format: Article
• Language: English
• Subjects: Permutations
• ISSN: 0942-5616; 1521-3870
• DOI: 10.1002/malq.202300031

Läuchli showed in the absence of the Axiom of Choice (AC$\mathsf {AC}$) that (2fin(m))ℵ0=2fin(m)$(2^{\textup {fin}(\mathfrak {m})})^{\aleph _0} = 2^{\textup {fin}(\mathfrak {m})}$ and, consequently, 22m+22m=22m$2^{2^{\mathfrak {m}}}+2^{2^{\mathfrak {m}}} = 2^{2^{\mathfrak {m}}}$ for all infinite cardinals m$\mathfrak {m}$, where fin(m)$\textup {fin}(\mathfrak {m})$ and 2m$2^{\mathfrak {m}}$ are the cardinalities of the set of finite subsets and the power set, respectively, of a set which is of cardinality m$\mathfrak {m}$. In this article, we give a generalisation of a simple form of Läuchli's lemma from which several results can be obtained. That is, 2m$2^{\mathfrak {m}}$ in the latter equation can be replaced by other cardinals which are equal to 2m$2^{\mathfrak {m}}$ in ZFC$\mathsf {ZF}{\rm C}$ but not in ZF$\mathsf {ZF}$, for example, m!$\mathfrak {m}!$ and Part(m)$\textup {Part}(\mathfrak {m})$, the cardinalities of the set of permutations and the set of partitions, respectively, of a set which is of cardinality m$\mathfrak {m}$.