페아노 공리

주세페 페아노가 제안한 자연수에 대한 공리.

1. $0$ is a natural number.
2. For every natural number $x$, $x = x$. That is, equality is reflexive.
3. For all natural numbers $x$ and $y$, if $x = y$, then $y = x$. That is, equality is symmetric.
4. For all natural numbers $x$, $y$ and $z$, if $x = y$ and $y = z$, then $x = z$. That is, equality is transitive.
5. For all $a$ and $b$, if $b$ is a natural number and $a = b$, then $a$ is also a natural number. That is, the natural numbers are closed under equality.
6. For every natural number $n$, $S(n)$ is a natural number. That is, the natural numbers are closed under $S$.
7. For all natural numbers $m$ and $n$, if $S(m)$ = $S(n)$, then $m$ = $n$. That is, $S$ is an injection.
8. For every natural number $n$, $S(n) = 0$ is false. That is, there is no natural number whose successor is $0$.
9. If $K$ is a set such that: $0$ is $K$, and for every natural number $n$, $n$ being in $K$ implies that $S(n)$ is in $K$, then $K$ contains every natural number.

See also

• en.wikipedia.org/wiki/Peano_axioms
• plato.stanford.edu/entries/truth-axiomatic/