Let us assume that the following four statements are Peano's first four axioms, leaving only the fifth (principle of mathematical induction).
1) Zero is a natural number. 2) Every natural number has a successor, which is also a natural number. 3) Zero is no successor to any natural number. 4) Two natural numbers that have equal successors are themselves equal.
Let s and t be successors of the natural number n. How can we prove that s = t? That is, how can we prove, using only the four axioms above, that every natural number has a single successor?
It seems to me that this is not possible. I need help.