Def/Natural number

 

=Peano Axioms= The natural numbers satisfy the Peano axioms, which are the following:
 * 1)  $0$ is a natural number.
 * 2)  If $n$ is a natural number, then its successor $Succ(n)$ is a natural number.
 * 3)  $0$ is not the successor of any natural number.
 * 4)  If $x$ and $y$ are natural numbers, and $Succ(x) = Succ(y)$, then $x = y$.
 * 5)  (Induction)  If $S \subset \NN$, and $0 \in S$, and $n \in S \Rightarrow Succ(n) \in S$, then $S = \NN$.

This is proven in the construction.

=Addition= There is a unique function $Add \colon \NN \times \NN \rightarrow \NN$, which satisfies the following three properties:
 * 1)  $Add(0,0) = 0$.
 * 2)  $Add(Succ(x), y) = Succ(Add(x,y))$.
 * 3)  $Add(x,Succ(y)) = Succ(Add(x,y))$.

From these properties, the following can be proven:
 * 1)  $Add(x,0) = x$. ($0$ is an additive identity)
 * 2)  $Add(x,y) = Add(y,x)$. (Addition is commutative)
 * 3)  $Add(x,Add(y,z)) = Add(Add(x,y), z)$.  (Addition is associative)

One typically writes $x + y$, instead of $Add(x,y)$. In this way, we find that: $$Succ(n) = Succ(Add(n,0)) = Add(n,Succ(0)) = Add(n,1) = n + 1.$$ In other words, the "successor" can be interpreted in the traditional way, as "adding 1".

=Multiplication= There is a unique function $Mult \colon \NN \times \NN \rightarrow \NN$, which satisfies the following properties:
 * 1)  $Mult(0,0) = 0$
 * 2)  $Mult(Succ(x), y) = Add(y, Mult(x,y))$.
 * 3)  $Mult(x, Succ(y)) = Add(Mult(x,y), x)$.

From these properties, the following can be proven:
 * 1)  $Mult(x,0) = 0$. ($0$ times anything equals zero)
 * 2)  $Mult(x,1) = x$. ($1$ is a multiplicative identity)
 * 3)  $Mult(x,y) = Mult(y,x)$. (Multiplication is commutative)
 * 4)  $Mult(x, Mult(y,z)) = Mult(Mult(x,y), z)$. (Multiplication is associative)
 * 5)  $Mult(x, Add(y,z)) = Add(Mult(x,y), Mult(x,z))$. (Multiplication distributes over addition)

One typically writes $x \cdot y$, or $x \times y$, or simply $xy$, instead of $Mult(x,y)$.

=Exponentiation= There is a unique function $Pow \colon \NN \times \NN \rightarrow \NN$, which satisfies the following properties:
 * 1)  $Pow(x,0) = 1$ for all $x \in \NN$ (we follow the somewhat unusual convention that $0^0 = 1$, within natural number exponentiation).
 * 2)  $Pow(x,Succ(y)) = Mult(x, Pow(x,y))$, for all $x,y \in \NN$.

From these properties, the following can be proven:
 * 1)  $Pow(Mult(x,y),z) = Mult(Pow(x,z), Pow(y,z))$.
 * 2)  $Pow(x,Mult(y,z)) = Pow(Pow(x,y),z)$.

One typically writes $x^y$, instead of $Pow(x,y)$.

=Metadata=