Peano Axioms

The Peano Axioms are a collection of certain truths on natural numbers that allow further development into complex numerical systems. Originally laid out by Giuseppe Peano, a mathematician who lived from 1858 to 1932, these axioms create a robust understanding of natural numbers. We'll go in order

Axiom 1: 0 is a natural number Axiom 2: If $n$ is a natural number, then $n$++ is also a natural number Axiom 3: 0 is not a successor of any natural number. $n$++ $\neq 0$ for any $n$. Axiom 4: Different natural numbers have different successors. If $n \neq m$, then $n$++ $\neq m$++ Axiom 5: Let $P(n)$ be any property pertaining to natural number $n$. Suppose that $P(0)$ is true, and suppose that whenever $P(n)$ is true, then $P(n$++$)$ is also true. Then $P(n)$ is true for every $n$.

These are, for the most part, all pretty straightforward. The last axiom here is interesting as it is the principle of mathematical induction, a very popular proof method and even relevant to field like Computer Science with techniques like dynamic programming.