CANTOR’S THEOREM MAY FAIL FOR FINITARY PARTITIONS
Guozhen Shen · Journal of Symbolic Logic · 2024
Abstract A partition is finitary if all its members are finite. For a set A , script upper B left parenthesis upper A right parenthesis $\mathscr {B}(A)$ B ( A ) denotes the set of all finitary partitions of A . It is shown consistent with upper Z upper F $\mathsf {ZF}$ Z F (without the axiom of choice) that there exist an infinite set A and a surjection from A onto script upper B left parenthesis upper A right parenthesis $\mathscr {B}(A)$ B ( A ) . On the other hand, we prove in upper Z upper F $\mathsf {ZF}$ Z F some theorems concerning script upper B left parenthesis upper A right parenthesis $\mathscr {B}(A)$ 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 script upper B left parenthesis upper A right parenthesis $\mathscr {B}(A)$ B ( A ) and no finite-to-one functions from script upper B left parenthesis upper A right parenthesis