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.

在\(\textbf{ZF}\)中，关于大小为\(|{\mathcal {P}} (X)|\)的X独立族的存在的著名 Fichtenholz–Kantorovich–Hausdorff 定理等同于同样著名的 Hewitt–Marczewski–Pondiczery 定理中关于乘积空间密度的以下部分：“乘积\({\textbf{2}}^{{\mathcal {P}}(X)}\)有大小的稠密子集 | X |”。然而，后一个陈述证明比\(\textbf{AC}\)严格弱，而完整的 Hewitt–Marczewski–Pondiczery 定理等同于\(\textbf{AC}\)。我们研究\(\textbf{ZF}\)在语句“X没有独立的大小族\(|{\mathcal {P}}(X)|\) ”并且 Levy 的经典论文中研究了“ X是有限的”的一些定义，观察到前一个陈述暗示了一个这样的定义, 是别人暗示的, 和别人无法比拟的。