Comment by LegionMammal978

Let X = "1 if ZF is consistent, 0 otherwise". Then the statements "X = 0" and "X = 1" are independent of ZF. Whether the definition of X is a satisfactory definition of a particular number is a question of mathematical philosophy.

BB(748) is very similar, in that I'd call it a 'definition' independent of ZF rather than a 'number' independent of ZF.