Mathematical logic and foundations Definition

Addition of natural numbers

Builds on Natural numbers Successor function Definition by recursion

Definition

Let $\mathbb{N}$ be the set of natural numbers and let $S$ be the successor function. Addition is the function $+ \colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ , $(a, b) \mapsto a + b$ , with these two properties for all $a$ and $b$ in $\mathbb{N}$ : $a + 0 = a, \qquad a + S(b) = S(a + b).$

Remarks

For each $a$ , these equations are a definition by recursion on $b$ . Thus exactly one such function exists. With $1 = S(0)$ , the second equation gives $a + 1 = S(a)$ . For example, $2 + 2 = S(2 + 1) = S(S(2 + 0)) = S(S(2)) = 4$ .

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Tag	`0W5DSDFQ4G`
URI	`m2://0W5DSDFQ4G`
Kind	definition
Writing	W0 stub
MSC	03F30
Defines	Addition of natural numbers
Uses	Natural numbers, Successor function, Definition by recursion

Addition of natural numbers

Definition

Remarks

Reading path

Where this leads

Further reading

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