Set $s$ is a natural number if $s$ is transitive and for every $x$, $y$ and $z$
- $y\in{s}\rightarrow(y$ is transitive$)$, and if
- $x\in{P}s\wedge(x$ is transitive$)\wedge{z}\in{P}x\wedge(z$ is transitive$)$ $\rightarrow{z}\in{x}\cup\{x\}$, and if
- $x\in{P}s\wedge(x$ is transitive$)\wedge\bigcup{x}\neq{x}$ $\rightarrow x=\bigcup{x}\cup\{\bigcup{x}\}$.
Then $\emptyset$ is the first natural number, the successor of natural number $s$ is defined by $s\cup\{s\}$ and the predecessor of number $s$ is defined by $\bigcup{s}$. All Peano axioms are straightforward proved, including the principle of mathematical induction. Also the natural number arithmetic is proved, for example if $a$ and $b$ are natural numbers then with induction is proved $a+b$ and $a\cdot{b}$ are natural numbers. The set $\omega$ of natural numbers is then postulated as $s\in\omega\leftrightarrow(s$ is a natural number).