You could just use 1 as the initial element though. Peano's axioms would become 1 != Sn Sn = Sm implies n=m n + 1 = Sn n + Sm = S(n + m) n

You could just use 1 as the initial element though. Peano's axioms would become 1 != Sn Sn = Sm implies n=m n + 1 = Sn n + Sm = S(n + m) n × 1 = n n × Sm = (n × m) + n Induction That's not any more complex than the usual set of axioms.