# Extended natural numbers

## Idea

The extended natural number system, denoted $\bar{\mathbb{N}}$, consists of all of the natural numbers together with an extra number representing infinity.

## Definition

Classically, $\bar{\mathbb{N}}$ is the disjoint union of the set $\mathbb{N}$ of natural numbers and a point $\{\infty\}$. That is,

$\bar{\mathbb{N}} = \{0,1,2,\ldots,\infty\} .$

In constructive mathematics, a more careful definition is required: an extended natural number is an infinite sequence $x$ of binary digits (each $0$ or $1$) with the property that $x_i = 1$ if $x_j = 1$ for any $j \leq i$; that is, the sequence is monotone. Then the natural number $n$ is identified with a sequence of $n$ copies of $0$ followed by $1$s, while infinity is identified with a sequence of all $0$s. It is not constructively valid that every natural number is either finite or infinite, but it is valid that any that is not finite is infinite, while Markov's principle is the converse.

## Universal property

$\bar{\mathbb{N}}$ comes naturally equipped with a map $pred\colon \bar{\mathbb{N}} \to 1 + \bar{\mathbb{N}}$ as defined below:

$pred(x) = \begin{cases} * & if\; x = 0 ,\\ n & if\; x = n + 1 ,\\ \infty & if\; x = \infty .\end{cases}$

Thus, it is a coalgebra for the endofunctor $H(X) = 1 + X$ on Set, and indeed is the terminal coalgebra for $H$. That is, given any set $S$ and map $p\colon S \to 1 + S$, there is a unique map $corec_S p\colon S \to \mathbb{N}$ such that

$\array { S & \stackrel{p}\to & 1 + S \\ \downarrow_{corec_S p} & & \downarrow_{\id_1 + corec_S p} \\ \bar{\mathbb{N}} & \stackrel{pred}\to & 1 + \bar{\mathbb{N}} }$

commutes. In this way, $\bar{\mathbb{N}}$ is dual to the natural number system? $\mathbb{N}$ in its guise as a natural numbers object.

You can think of $corec_S p$ as mapping an element $a$ of $S$ to the number of times that $p$ must be applied in succession, starting from $a$, before being taken out of $S$. Since this may never occur, we need $\infty$ as a possible value.

Note that this universal property also holds constructively (which is why we can be sure that the constructive definition above is correct). We define $pred$ constructively as follows:

$pred(x) = \begin{cases} * & if\; x_0 = 1 ,\\ (x_1,x_2,\ldots) & if\; x_0 = 0 .\end{cases}$

## Topology

We may naturally give $\bar{\mathbb{N}}$ a topology giving it the structure of a compact Hausdorff space; unusually, this works even in weak constructive foundations (without having to use the fan theorem or pass to a locale).

We may define the topology simply (and constructively) as follows: a subset $G$ of $\bar{\mathbb{N}}$ is open if, whenever $\infty \in G$, there is a finite $n$ such that $m \in G$ whenever $m \geq n$. In other words, $G$ is a neighbourhood of $\infty$ just when almost every finite number also belongs to $G$.

In this way, $\bar{\mathbb{N}}$ is the Alexandroff compactification of the discrete space $\mathbb{N}$. The space is obviously compact because, given an open cover $\mathcal{U}$, we have $\infty \in G \in \mathcal{U}$ for some $G$, so $G$ alone contains almost every point, and only finitely many more open sets are needed.

It is sometimes convenient to represent $\bar{\mathbb{N}}$ as a subspace of the real line $\mathbb{R}$, which we can do by interpreting the natural number $n$ as $2^{-n}$ and $\infty$ as $0$. Constructively, the monotone bit sequence $x$ becomes the real number

$\frac{1}{2} {\sum_{i=0}^\infty x_i 2^{-i}} ,$

which always converges. Another common representation uses $1/(n+1)$ instead of $1/2^n$.

Given any topological space $X$, an infinite sequence in $X$ may be thought of as a continuous map to $X$ from the discrete space $\mathbb{N}$. Then this sequence converges? iff this map can be extended to a continuous map on $\bar{\mathbb{N}}$. For this reason, $\bar{\mathbb{N}}$ is sometimes called the universal convergent sequence. (Strictly speaking, unless $X$ is at least sequentially Hausdorff, a map to $X$ from $\bar{\mathbb{N}}$ contains more information than a sequence in $X$ with the property of convergence.)

This may all be generalised from sequences to other nets; given a directed set $D$, we form $\bar{D}$ by adjoining $\infty$ and taking $G \subseteq D$ as a neighbourhood of $\infty$ iff $G$ owns almost all of $D$. (Constructively, this may require using locales for the general case.)

## In higher category theory

Concepts in higher category theory often come in $n$-versions where $n$ is an extended natural number. (Sometimes it’s also possible to give $n$ a few negative values as well; see negative thinking.) Typically, the $\infty$-version is all-encompassing, with the $n$-versions as special cases. On the other hand, the $1$-version is usually more familiar outside of category theory.

The claim that every extended natural number is either finite or infinite is equivalent to the limited principle of omniscience ($LPO$) for natural numbers. On the other hand, the $LPO$ for extended natural numbers is simply true; given any function from $\bar{\mathbb{N}}$ to $\{0,1\}$, it is either all $0$s or has a $1$. See Escardó (2011).