Mathematical logic and foundations Definition

Order on the natural numbers

Builds on Natural numbers Addition of natural numbers

Definition

Let $a$ and $b$ be natural numbers. The notation $a \le b$ means that there is a natural number $c$ with $a + c = b$ , where $+$ is the addition of $\mathbb{N}$ . The notation $a < b$ means that $a \le b$ and $a \neq b$ .

Remarks

The relation $\le$ is a total order on $\mathbb{N}$ : it is reflexive, antisymmetric and transitive, and $a \le b$ or $b \le a$ for all $a$ and $b$ . The order agrees with the operations: if $a \le b$ , then $a + c \le b + c$ and $a c \le b c$ for each natural number $c$ . Each nonempty set of natural numbers has a least element for this order. This is the well-ordering principle.

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Tag	`0Y9H52BT03`
URI	`m2://0Y9H52BT03`
Kind	definition
Writing	W0 stub
MSC	03F30
Defines	Order on the natural numbers
Uses	Natural numbers, Addition of natural numbers

Order on the natural numbers

Definition

Remarks

Reading path

Where this leads

Further reading

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