A choice-free cardinal equality (Q2075277)

If \(x,y\) are arbitrary sets, then \(x^{y}\), \([x]^{y}\), and \(\mathrm{fin}(x)\) denote the set of all functions from \(y\) to \(x\), the set of all subsets of \(x\) which are equipotent to \(y\), and the set of all finite subsets of \(x\), respectively. The author of the paper under review proves in \(\mathsf{ZF}\) (i.e., Zermelo-Fraenkel set theory minus the axiom of choice (\(\mathsf{AC}\))) that, for all infinite sets \(x\) and all natural numbers \(n\), \[ |\wp(\mathrm{fin}(x)^{n})|=|\wp([\mathrm{fin}(x)]^{n})|, \] i.e., there is a bijection \(f:\wp(\mathrm{fin}(x)^{n})\rightarrow\wp([\mathrm{fin}(x)]^{n})\). (As noted in the paper, it is known that \(\mathrm{fin}(x)^{n}\) and \([\mathrm{fin}(x)]^{n}\) need not be equipotent in the absence of \(\mathsf{AC}\).) He also proves that the following statement is relatively consistent with \(\mathsf{ZF}\): there exists an infinite set \(a\) such that \[ |\wp(\mathrm{fin}(a))|<|\wp(\mathrm{fin}(a)^{2})|<|\wp(\mathrm{fin}(a)^{3})|<\cdots<|\wp(\mathrm{fin}(\mathrm{fin}(a)))|, \] where, for sets \(x\) and \(y\), \(|x|<|y|\) means that there is an injection from \(x\) into \(y\), but there is no injection from \(y\) into \(x\). To establish this result, the author first shows that the above statement holds true for the infinite set \(A\) of atoms of the basic Fraenkel permutation model (Lemma 4.1 of the paper), and then uses the Jech-Sochor first embedding theorem to transfer the result to \(\mathsf{ZF}\) (Corollary 4.2 of the paper). Furthermore, it is shown that, for the set \(A\) of atoms of the basic Fraenkel model and for every \(n\in\omega\), \(\mathrm{fin}(A)^{n}\) is dually Dedekind finite in the model, that is, there is no noninjective surjection from \(\mathrm{fin}(A)^{n}\) onto \(\mathrm{fin}(A)^{n}\) (Lemma 4.3 of the paper), and the result is transferred to \(\mathsf{ZF}\) via the Jech-Sochor theorem. The paper is concluded with the following three open questions: \begin{itemize} \item[1.] Does \(\mathsf{ZF}\) prove Lemmas 4.1 and 4.3 for all amorphous sets, or strongly amorphous sets? (An amorphous set is an infinite set with no partition into two infinite sets. An amorphous set is strongly amorphous if every partition of it has only finitely many nonsingletons.) \item[2.] Does \(\mathsf{ZF}\) prove that \(|\wp(\wp(\mathrm{fin}(x)))|=|\wp(\wp(\mathrm{fin}(\mathrm{fin}(x))))|\) for any infinite set \(x\)? \item[3.] Does \(\mathsf{ZF}\) prove that \(|\wp(\wp(x))|=|\wp(\wp(x\cup\{a\}))|\) for any infinite set \(x\) and any \(a\not\in x\)? \end{itemize} Question 3 was also asked by \textit{H. Läuchli} [Z. Math. Logik Grundlagen Math. 7, 141--145 (1961; Zbl 0114.01005)], who proved in \(\mathsf{ZF}\) that \(|\wp(\wp(x))|=|\wp(\wp(x)\cup\{a\})|\) for all infinite sets \(x\) and all \(a\not\in\wp(x)\).