In ZF set theory without the axiom of choice, the following definitions for finiteness of a set "S" are distinct, and are arranged in strictly decreasing order of strength.
The following definitions for finiteness of a set "S" are distinct. This is in ZF. This ZF is set theory without the axiom of choice. The following definitions for finiteness of a set "S" are arranged in strictly decreasing order of strength. This was in ZF. This ZF was set theory without the axiom of choice.