Sep 14th 2011, 17:29:06
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?
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.
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.
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.