Definition:Multiplication/Natural Numbers

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Definition

Let $\N$ be the natural numbers.

Multiplication on $\N$ is the basic operation $\times$ everyone is familiar with.

For example:

$3 \times 4 = 12$
$13 \times 7 = 91$


Every attempt to describe the natural numbers via suitable axioms should reproduce the intuitive behaviour of $\times$.

The same holds for any construction of $\N$ in an ambient theory.


Multiplication in terms of Addition

Let $+$ denote addition.


The binary operation $\times$ is recursively defined on $\N$ as follows:

$\forall m, n \in \N: \begin{cases}

m \times 0 & = 0 \\ m \times \paren {n + 1} & = m \times n + m \end{cases}$

This operation is called multiplication.


Equivalently, multiplication can be defined as:

$\forall m, n \in \N: m \times n := \mathop {+^n} m$

where $\mathop {+^n} m$ denotes the $n$th power of $m$ under $+$.


$1$-Based Natural Numbers

Let $\N_{>0}$ be the $1$-based natural numbers, axiomatized by:


\((\text A)\)   $:$     \(\ds \exists_1 1 \in \N_{> 0}:\) \(\ds a \times 1 = a = 1 \times a \)      
\((\text B)\)   $:$     \(\ds \forall a, b \in \N_{> 0}:\) \(\ds a \times \paren {b + 1} = \paren {a \times b} + a \)      
\((\text C)\)   $:$     \(\ds \forall a, b \in \N_{> 0}:\) \(\ds a + \paren {b + 1} = \paren {a + b} + 1 \)      
\((\text D)\)   $:$     \(\ds \forall a \in \N_{> 0}, a \ne 1:\) \(\ds \exists_1 b \in \N_{> 0}: a = b + 1 \)      
\((\text E)\)   $:$     \(\ds \forall a, b \in \N_{> 0}:\) \(\ds \)Exactly one of these three holds:\( \)      
\(\ds a = b \lor \paren {\exists x \in \N_{> 0}: a + x = b} \lor \paren {\exists y \in \N_{> 0}: a = b + y} \)      
\((\text F)\)   $:$     \(\ds \forall A \subseteq \N_{> 0}:\) \(\ds \paren {1 \in A \land \paren {z \in A \implies z + 1 \in A} } \implies A = \N_{> 0} \)      


The operation $\times$ in this axiomatization is called multiplication.


Also defined as

Some treatments of Peano's axioms define the non-successor element (or primal element) to be $1$ and not $0$.

The treatments are similar, but the $1$-based system results in an algebraic structure which has no identity element for addition, and so no zero for multiplication.


Under this $1$-based system, multiplication is consequently defined as follows:

$\forall m, n \in P: \begin{cases}

m \times 1 & = m \\ m \times \map s n & = m \times n + n \end{cases}$

or:

$\forall m, n \in P: \begin{cases}

1 \times n & = n \\ \map s m + n & = m \times n + n \end{cases}$


Also see

  • Results about natural number multiplication can be found here.


Sources

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