Because it's the same size. I can show it's the same size because I can just change the labels on the sets (label change of elements doesn't change group size)

Because it's the same size. I can show it's the same size because I can just change the labels on the sets (label change of elements doesn't change group size) and go back and forth. In this case, 0 is the sandwich, and all the nonnegative numbers get relabelled down by one. Going back, sandwiches become 0 and you add one to nonnegative numbers. So they've got to be the same size, if I can go back and forth and everything just gets a new label and I don't miss anything.