## The Axiom of Proof

Mathematical Induction is not just a technique but a fundamental method of proof for the natural numbers.  The natural numbers are 0,1,2, and so on.  The axiom of induction is what handles the and so on.

Axiom of Induction

If zero is a member of a particular set of natural numbers, and if the successor of any member is likewise a member, then said set of natural numbers is the set of all natural numbers.

The axiom of induction is not beyond the comprehension of children in elementary school.  It is the succession of days, the generations, and many other concepts they learn.

The Axiom of Induction is fundamental to proving the properties of the natural numbers.  It is also necessary to define addition and multiplication.  This takes some additional work in the form of the recursion theorem.  Is is this additional work that requires set theory, ordered pairs, and functions.  That was what Dedekind realized in his 1888 book, “Was Sind und Was Sollen die Zahlen.”  Literally, this translates as “What are and what should be the numbers.”

To define addition and prove it is commutative and associative, we must use the Axiom of Inductive Proof as we might also call it.  This is an essential point of mathematics.  We take away the foundation of math when we don’t teach mathematical induction.  Works alone can not lead to math understanding, but proof and works together.  The Axiom of Inductive Proof is the core of proof of the properties of the natural numbers.  We can not abandon it.