The Real Numbers and Standard Notation

The real numbers are a Dedekind complete ordered field, denoted $\mathbb{R}$, with order relation $\le$.

We use the following notation on $\mathbb{R}$. Let $a,b\in\mathbb{R}$.

1. (Arithmetic notation) $0$ and $1$ denote the additive and multiplicative identity elements of $\mathbb{R}$. For $x\in\mathbb{R}$, $-x$ denotes the additive inverse of $x$ and, when $x\ne 0$, $x^{-1}$ denotes the multiplicative inverse of $x$, all written as in the field axioms. We write

2. (Order notation) $<$ denotes the strict order associated with $\le$; we write $a\ge b$ for $b\le a$, and $a>b$ for $b<a$. We call $a$ nonnegative if $0\le a$, positive if $0<a$, nonpositive if $a\le 0$, and negative if $a<0$.

3. (Natural numbers in $\mathbb{R}$) $\iota_{\mathbb{R}}$ denotes the canonical map into $\mathbb{R}$ from the natural numbers $\mathbb{N}$. For $n\in\mathbb{N}$ other than $1$, we also write $n$ for the real number $\iota_{\mathbb{R}}(n)$ wherever an element of $\mathbb{R}$ is required; the symbol $1$ retains its meaning from clause 1.