---
tag: 066RJJVWH9
kind: statement
role: proposition
name: The integers are countable
msc: ["03E10"]
uses: [0SEZ3JGZP7, 0SNQEF7Z53]
sources:
  - url: https://en.wikipedia.org/wiki/Countable_set
    locator: Introduction
---

## Statement

The set $\mathbb{Z}$ of integers is [[countably infinite|0SEZ3JGZP7]].
The function $f \colon \mathbb{N} \to \mathbb{Z}$ with $f(2k) = k$ and $f(2k + 1) = -(k + 1)$ is a [[bijection|0SNQEF7Z53]].

## Remarks

The bijection lists the integers as $0, -1, 1, -2, 2, \dots$.
So $|\mathbb{Z}| = |\mathbb{N}| = \aleph_0$, although $\mathbb{N}$ is a proper subset of $\mathbb{Z}$.

## Explanations

### L1 human

The list $0, 1, 2, \dots$ never reaches the negative integers.
Alternate between the two signs, and every integer gets a position in the list.