# Category:Definitions/Peano's Axioms

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This category contains definitions related to Peano's Axioms.

Related results can be found in Category:Peano's Axioms.

**Peano's Axioms** are intended to reflect the intuition behind $\N$, the mapping $s: \N \to \N: \map s n = n + 1$ and $0$ as an element of $\N$.

Let there be given a set $P$, a mapping $s: P \to P$, and a distinguished element $0$.

Historically, the existence of $s$ and the existence of $0$ were considered the first two of **Peano's Axioms**:

\((\text P 1)\) | $:$ | \(\displaystyle 0 \in P \) | $0$ is an element of $P$ | |||||

\((\text P 2)\) | $:$ | \(\displaystyle \forall n \in P:\) | \(\displaystyle \map s n \in P \) | For all $n \in P$, its successor $\map s n$ is also in $P$ |

The other three are as follows:

\((\text P 3)\) | $:$ | \(\displaystyle \forall m, n \in P:\) | \(\displaystyle \map s m = \map s n \implies m = n \) | $s$ is injective | ||||

\((\text P 4)\) | $:$ | \(\displaystyle \forall n \in P:\) | \(\displaystyle \map s n \ne 0 \) | $0$ is not in the image of $s$ | ||||

\((\text P 5)\) | $:$ | \(\displaystyle \forall A \subseteq P:\) | \(\displaystyle \paren {0 \in A \land \paren {\forall z \in A: \map s z \in A} } \implies A = P \) | Principle of Mathematical Induction: | ||||

Any subset $A$ of $P$, containing $0$ and | ||||||||

closed under $s$, is equal to $P$ |

## Pages in category "Definitions/Peano's Axioms"

The following 4 pages are in this category, out of 4 total.