Axiom:Peano's Axioms/Formulation 1
== 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''': {{begin-axiom}} {{axiom | n = \text P 1         | q =          | m = 0 \in P         | t = $0$ is an element of $P$ }} {{axiom | n = \text P 2         | q = \forall n \in P         | m = \map s n \in P         | t = For all $n \in P$, its successor $\map s n$ is also in $P$ }} {{end-axiom}} The other three are as follows:  {{begin-axiom}} {{axiom | n = \text P 3         | q = \forall m, n \in P         | m = \map s m = \map s n \implies m = n         | t = $s$ is injective }} {{axiom | n = \text P 4         | q = \forall n \in P         | m = \map s n \ne 0         | t = $0$ is not in the image of $s$ }} {{axiom | n = \text P 5         | q = \forall A \subseteq P         | m = \paren {0 \in A \land \paren {\forall z \in A: \map s z \in A} } \implies A = P         | t = Principle of Mathematical Induction: }} {{axiom | t = Any subset $A$ of $P$, containing $0$ and }} {{axiom | t = closed under $s$, is equal to $P$ }} {{end-axiom}}  == Also defined as == {{:Axiom:Peano's Axioms/Also defined as}} == Also known as == {{:Axiom:Peano's Axioms/Also known as}} == Also see == * Equivalence of Formulations of Peano's Axioms {{LinkToCategory|Peano's Axioms|Peano's axioms}} {{NamedforAxiom|Giuseppe Peano}} == Historical Note == {{:Axiom:Peano's Axioms/Historical Note}} == Sources == * {{BookReference|Naive Set Theory|1960|Paul R. Halmos|prev = Definition:Transitive Set|next = Axiom:Peano's Axioms/Historical Note}}: $\S 12$: The Peano Axioms * {{BookReference|Number Theory|1964|J. Hunter|prev = Definition:Successor Mapping|next = Axiom:Peano's Axioms/Also defined as}}: Chapter $\text {I}$: Number Systems and Algebraic Structures: $2$. The positive integers * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Mathematician:Giuseppe Peano|next = Definition:Successor Mapping}}: $\S 16$ * {{BookReference|A Handbook of Terms used in Algebra and Analysis|1972|A.G. Howson|prev = Definition:Natural Numbers|next = Principle of Finite Induction}}: $\S 4$: Number systems $\text{I}$: Peano's Axioms * {{BookReference|Introduction to Topology|1975|Bert Mendelson|ed = 3rd|edpage = Third Edition|prev = Definition:Natural Numbers|next = Principle of Mathematical Induction for Peano Structure}}: Chapter $1$: Theory of Sets: $\S 1$: Introduction * {{BookReference|Numbers, Sets and Axioms|1982|Alan G. Hamilton|prev = Natural Number Multiplication is Cancellable|next = Axiom:Peano's Axioms/Historical Note}}: $\S 1$: Numbers: $1.1$ Natural Numbers and Integers * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Peano Curve|next = Mathematician:Karl Pearson|entry = Peano's postulates}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Peano Curve|next = Mathematician:Karl Pearson|entry = Peano's postulates}} * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Basic Universe|next = Axiom:Peano's Axioms/Historical Note}}: Chapter $3$: The Natural Numbers: $\S 1$ Preliminaries * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Axiom:Peano's Axioms/Historical Note|next = Definition:Peano Curve|entry = Peano's axioms}}