Independent families and some notions of finiteness

In \(\textbf{ZF}\), the well-known Fichtenholz–Kantorovich–Hausdorff theorem concerning the existence of independent families of X of size \(|{\mathcal {P}} (X)|\) is equivalent to the following portion of the equally well-known Hewitt–Marczewski–Pondiczery theorem concerning the density of product spaces: “The product \({\textbf{2}}^{{\mathcal {P}}(X)}\) has a dense subset of size |X|”. However, the latter statement turns out to be strictly weaker than \(\textbf{AC}\) while the full Hewitt–Marczewski–Pondiczery theorem is equivalent to \(\textbf{AC}\). We study the relative strengths in \(\textbf{ZF}\) between the statement “X has no independent family of size \(|{\mathcal {P}}(X)|\)” and some of the definitions of “X is finite” studied in Levy’s classic paper, observing that the former statement implies one such definition, is implied by another, and incomparable with some others.