martian - every element other than 0 is a successor is true if 0 and its successors are the entire set. But what if 0 and its successors are not the entire set?

The way you phrased the induction axiom seems to imply that, but the way I've seen the induction axiom phrased does not seem to imply that. I guess there is variance in the way the induction axiom is phrased, but it seems odd for such a large variance to exist.

additionally, 0 cannot be the successor of itself, because 0 is not the successor of any number. So {0,0,0,0...} is not valid.