We know that, with Axiom of Choice (**AC**), it can be shown that there exists a finitely additive uniform distribution defined for all subsets of the integers (see, e.g., Hrbacek and Jech 1999, Ch. 11). 

I was wondering, WITHOUT **AC**, whether or not there exists any finitely additive measure defined over $\mathcal{P}(\mathbb{N})$? Is there a result showing that **ZF** + {nonexistence of finitely additive measure over  $\mathcal{P}(\mathbb{N})$} is consistent? Thanks!