I think you mean "I think you mean all finite subsets of natural numbers" ;)

The power set of the natural numbers is the union of the finite subsets of the natural numbers and the infinite subsets of the natural numbers. The former is in bijective correspondence with N.* So if the infinite subset were in bijective correspondence with N, then N would be in bijective correspondence with its powerset. But this is not the case by Cantor's theorem, so there cannot be a bijection between N and its infinite subsets.

*Using:
{}->[empty sum]+1=1
{1}->2^(1-1)+1=2
{2}->2^(2-1)+1=3
{1,2}->2^(1-1)+2^(2-1)+1=1+2+1=4
{2,3,5,7}->2^(2-1)+2^(3-1)+2^(5-1)+2^(7-1)+1=2+4+16+64+1 etc.

Also mentioned at http://lesswrong.com/lw/j8/the_crackpot_offer/