Proof of The Integers and the Rational Numbers are Countable

Throughout, $\mathbb{N}$ is the set of natural numbers, $\iota:\mathbb{N}\to\mathbb{R}$ is the canonical map of $\mathbb{R}$, claim numbers for $\iota$ refer to Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and claim numbers for $\mathbb{Z}$ refer to Arithmetic, Order, Discreteness and Intervals of the Integers. For $s,t\in\mathbb{R}$, $s-t$ abbreviates $s+(-t)$ in the ordered field $\mathbb{R}$.

Claim 1. Define

$$g:\mathbb{N}\times\mathbb{N}\to\mathbb{R},\qquad g(m,n)=\iota(m)-\iota(n).$$

Each value of $g$ is an integer: $\iota(m),\iota(n)\in\mathbb{Z}$ by the definition of $\mathbb{Z}$, and $\mathbb{Z}$ is closed under the difference of two of its elements by claim 2.

Conversely, every integer is a value of $g$. By claim 1, $\iota(1)=1$ and $\iota(n+1)=\iota(n)+1$ for $n\in\mathbb{N}$. Hence

$$g(1,1)=1-1=0,\qquad g(n+1,1)=\bigl(\iota(n)+1\bigr)-1=\iota(n),\qquad g(1,n+1)=1-\bigl(\iota(n)+1\bigr)=-\iota(n)$$

for every $n\in\mathbb{N}$. By the definition of $\mathbb{Z}$, every integer is $0$, or $\iota(n)$, or $-\iota(n)$ for some $n\in\mathbb{N}$, so $\mathbb{Z}=\{g(w):w\in\mathbb{N}\times\mathbb{N}\}$. Since $\mathbb{N}\times\mathbb{N}$ is countable by The Set of Pairs of Natural Numbers is Countable, claim 4 of Basic Properties of Countable Sets shows that $\mathbb{Z}$ is countable.

Claim 2. The set $\mathbb{Z}\setminus\{0\}$ is a subset of $\mathbb{Z}$, hence countable by claim 1 above and claim 3 of Basic Properties of Countable Sets. By claim 1 of Products and Powers of Countable Sets, the Cartesian product $\mathbb{Z}\times(\mathbb{Z}\setminus\{0\})$ is countable. Define

$$q:\mathbb{Z}\times(\mathbb{Z}\setminus\{0\})\to\mathbb{R},\qquad q(a,b)=a\,b^{-1},$$

which is well defined because $b\ne0$ has a multiplicative inverse in the field $\mathbb{R}$. By the definition of $\mathbb{Q}$, the set of values of $q$ is exactly $\mathbb{Q}$, so claim 4 of Basic Properties of Countable Sets shows that $\mathbb{Q}$ is countable.

Claim 3. Immediate from claim 2 and claim 2 of Products and Powers of Countable Sets, applied with $X=\mathbb{Q}$.