peano axioms

if $\text{[math]}$ is a natural number, then $\text{[math]}$ (its increment) is also a natural number.

0 is not the successor of any natural number; i.e., we have $\text{[math]}$ for every natural number $\text{[math]}$ .

different natural numbers must have different successors: i.e., if $\text{[math]}$ are natural numbers and $\text{[math]}$ , then $\text{[math]}$ . equivalently, if $\text{[math]}$ then we must have $\text{[math]}$ .

let $\text{[math]}$ be any property pertaining to a natural number $\text{[math]}$ . suppose that $\text{[math]}$ is true, and suppose that whenever $\text{[math]}$ is true, $\text{[math]}$ is also true. then $\text{[math]}$ is true for every natural number $\text{[math]}$ .

[cite:@tao_analysis_1 chapter 2 definition 2.1.1]