I'm not sure in what sense there isn't a recipe for arms in our DNA. To me, it seems the DNA does encode that stuff, but in a highly compressed format that is then "unzipped" through the laws of physics and biology into a living and breathing being with arms.

I mean, the information has to be in there somewhere, right?

I don't know either, maybe epigenetics play a part in this (Some information transferred from the mother cells to the child)?