Cantor’s Theorem May Fail for Finitary Partitions

A partition is finitary if all its members are finite. For a set A , B(A) denotes the set of all finitary partitions of A . It is shown consistent with ZF (without the axiom of choice) that there exist an infinite set A and a surjection from A onto B(A) . On the other hand, we prove in ZF some theorems concerning B(A) for infinite sets A , among which are the following: (1) If there is a finitary partition of A without singleton blocks, then there are no surjections from A onto B(A) and no finite-to-one functions from B(A) to A . (2) For all n∈ ω , |Aⁿ|<|B(A)| . (3) |B(A)|≠ |seq(A)| , where seq(A) is the set of all finite sequences of elements of A .