I've been studying the construction of the natural numbers, and I can't solve my own question, namely
Why is it necessary to use mathematical induction?
Let me clarify this. For example, we know that, for all $n,m \in \mathbb{N}$, then $n \cdot m = m \cdot n$. In order to prove this, we use mathematical induction, but when we think about $n,m \in \mathbb{R}$ (real numbers), in order to prove $n \cdot m= m \cdot n$, we don't need mathematical induction.
Why sometimes in a proof it's enough to take $x \in \mathbb{R}$, arbitrary number, but for natural numbers mathematical induction is necessary?
Anyone can help me, please?
I hope somebody can give me a hint in order to understand this question.
Best Answer
There are two ways to introduce the real numbers.
First way: a set of layers, starting from the natural numbers, then the integers, then the rational numbers, then the real numbers.
Second way: a set of axioms for the real numbers is taken and a structure satisfying them is assumed to exist.
In the first way, we need to prove commutativity of multiplication in the integers, then in the rational numbers, and then in the real numbers. This chain of proofs is based on the proof of commutativity of multiplication in the natural numbers.
In the second way, commutativity of multiplication is taken as an axiom. But we need to embed the natural numbers in the real numbers and this again requires induction and commutativity of multiplication in the natural numbers.