Category:Axioms/Peano's Axioms
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This category contains axioms related to 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:
\((\text P 1)\) | $:$ | \(\ds 0 \in P \) | $0$ is an element of $P$ | ||||||
\((\text P 2)\) | $:$ | \(\ds \forall n \in P:\) | \(\ds \map s n \in P \) | For all $n \in P$, its successor $\map s n$ is also in $P$ |
The other three are as follows:
\((\text P 3)\) | $:$ | \(\ds \forall m, n \in P:\) | \(\ds \map s m = \map s n \implies m = n \) | $s$ is injective | |||||
\((\text P 4)\) | $:$ | \(\ds \forall n \in P:\) | \(\ds \map s n \ne 0 \) | $0$ is not in the image of $s$ | |||||
\((\text P 5)\) | $:$ | \(\ds \forall A \subseteq P:\) | \(\ds \paren {0 \in A \land \paren {\forall z \in A: \map s z \in A} } \implies A = P \) | Principle of Mathematical Induction: | |||||
Any subset $A$ of $P$, containing $0$ and | |||||||||
closed under $s$, is equal to $P$ |
Pages in category "Axioms/Peano's Axioms"
The following 5 pages are in this category, out of 5 total.