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Informally, the positive integers are the set:

$\Z_{\ge 0} = \set {0, 1, 2, 3, \ldots}$

As the set of integers $\Z$ is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same.

Thus positive can be formally defined on $\Z$ as a relation induced on those equivalence classes as specified in the definition of integers.

That is, the integers being defined as all the difference congruence classes, positive can be defined directly as the relation specified as follows:

The integer $z \in \Z: z = \eqclass {\tuple {a, b} } \boxminus$ is positive if and only if $b \le a$.

The set of positive integers is denoted $\Z_{\ge 0}$.

An element of $\Z$ can be specifically indicated as being positive by prepending a $+$ sign:

$+x := x \in \Z_{\ge 0}$.

Ordering on Integers

Definition 1

The integers are ordered on the relation $\le$ as follows:

$\forall x, y \in \Z: x \le y$

if and only if:

$\exists c \in P: x + c = y$

where $P$ is the set of positive integers.

That is, $x$ is less than or equal to $y$ if and only if $y - x$ is non-negative.

Definition 2

The integers are ordered on the relation $\le$ as follows:

Let $x$ and $y$ be defined as from the formal definition of integers:

$x = \eqclass {x_1, x_2} {}$ and $y = \eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \in \N$.


$x < y \iff x_1 + y_2 \le x_2 + y_1$


$+$ denotes natural number addition
$\le$ denotes natural number ordering.

Also known as

As there is often confusion as to whether or not $0$ is included in the set of positive integers, it may be preferable to refer to the set of non-negative integers instead.

The notation $\Z^+$ is common, but leaves it ambiguous as to whether $\Z_{>0}$ or $\Z_{\ge 0}$ is meant.

Also see