Properties of the Canonical Map from the Natural Numbers to an Ordered Field

Let $F$ be an ordered field, with the addition, multiplication, additive identity $0$, multiplicative identity $1$ and multiplicative inverses $a^{-1}$ of the underlying field, and with its order $\le$; for $a,b\in F$ write $a<b$ to mean $a\le b$ and $a\ne b$. Let $\mathbb{N}$ be the set of natural numbers, with addition and multiplication as in that definition and with the order $\le$, and let $\iota_{F}:\mathbb{N}\to F$ be the canonical map of $F$.

Then the following hold for all $m,n\in\mathbb{N}$.

1. (Base and step)¶ $\iota_{F}(1)=1$ and $\iota_{F}(n+1)=\iota_{F}(n)+1$.

2. (The unit is a lower bound)¶ $1\le\iota_{F}(n)$.

3. (Positivity)¶ $0<\iota_{F}(n)$; consequently $\iota_{F}(n)\ne 0$, the inverse $\iota_{F}(n)^{-1}$ exists, and $0<\iota_{F}(n)^{-1}$.

4. (Additivity)¶ $\iota_{F}(m+n)=\iota_{F}(m)+\iota_{F}(n)$.

5. (Multiplicativity)¶ $\iota_{F}(mn)=\iota_{F}(m)\,\iota_{F}(n)$.

6. (Strict monotonicity)¶ If $m<n$ in $\mathbb{N}$, then $\iota_{F}(m)<\iota_{F}(n)$ in $F$.

7. (Injectivity)¶ If $\iota_{F}(m)=\iota_{F}(n)$, then $m=n$.