# Peano axiom of induction- why cannot it be replaced by something simpler?

The Peano axiom of induction for natural numbers says that For any property $$P(n)$$ , if $$P(0)$$ holds, and that whenever $$P(n)$$ holds, $$P(n++)$$ holds, then $$P(n)$$ holds for all natural numbers.

Can this be replaced by

"For every natural number $$n$$ not equal to $$0$$, there exists another natural number $$m$$ such that $$m++ = n$$" ?

What are the problems associated with replacing this?

I am following Terence Tao's book Analysis I.

• The axiom of induction offers a principle for proving some statement $P$ for the natural numbers. Your "axiom" does not even refer to some property $P$ to be proven over the natural numbers... How are the two connected? Mar 17 '19 at 10:04
• One of the reasons mentioned in the book to use the axiom of induction was to remove superfluous/extra elements. For example, the set Z = (0, 0++, 0++++, ........, x, x++, x++++.....) follows the previous axioms (where x is not a natural number). We could eliminate x and its increments by what I wrote in the suggestion. What I want to know is how does the induction axiom add more than just removing superfluous elements. Mar 18 '19 at 10:13

Let $$N=\{-1\}\times\mathbb N\cup\{1\}\times\mathbb Z$$. If $$(\pm1,m)\in N$$, then let $$(\pm1,m)++=(\pm1,m+1)$$. Then all the Peano axioms (with the induction one replaced by yours) hold (assuming that $$0=(-1,0)$$), but what we have here is something which is different from the naturals. For instance, induction doesn't hold here.