User:Dfeuer/Peano Axioms

Axiom

NOTE: I am well aware that these are phrased differently from the ones in Main. We can resolve the differences later—nothing horrible here.

\((P_1)\)	$:$							$0$ is a natural number.
\((P_2)\)	$:$							If $n$ is a natural number, so is $n^+$.
\((P_3)\)	$:$							For any natural number $n$, $n^+ ≠ 0$.
\((P_4)\)	$:$							For any natural numbers $m$ and $n$, if $m^+ = n^+$ then $m = n$
\((P_5)\)	$:$							Principle of Mathematical Induction For any set $A$, if the following conditions hold then $A$ contains every natural number: $(1)\quad 0 \in A$ $(2)\quad$For every natural number $n$, $n \in A \implies n^+ \in A$.