The real numbers, when constructed as Dedekind cuts over the rationals, form a field.

We shall often write the one-sided Dedekind cut $(A,B)$ ^[1] as simply $A$ (using bold face for Dedekind cuts); we can do this because if we already know $A$ then $B$ is completely determined. This will make our notation less messy.

The field structure, together with the total ordering on it, is as follows (where we write $0$ for the Dedekind cut $\left(\right\{r\in Q\mid r<0\},\{r\in Q\mid r\ge 0\left\}\right)$):

• $(A,B)+(C,D)=(A+C,B+D)$
• $A\le C$ if and only if everything in $A$ lies in $C$.
• Multiplication is somewhat complicated.
  • If $0\le A$, then $A\times C=\{ac\mid a\in A,a>0,c\in C\}$.
  • If $A<0$ and $0\le C$, then $A\times C=\{ac\mid a\in A,c\in C,c>0\}$.
  • If $A<0$ and $C<0$, then $A\times C=\left\{\right\}$

where $(A,B)$ is a one-sided Dedekind cut (so that $A$ has no greatest element).

(Here, the "set sum" $A+C$ is defined as "everything that can be made by adding one thing from $A$ to one thing from $C$": namely, $\{a+c\mid a\in A,c\in C\}$ in set builder notation; and $A\times C$ is similarly $\{a\times c\mid a\in A,c\in C\}$.)

Proof

Well-definedness

We need to show firstly that these operations do in fact produce Dedekind cuts.

Addition

Firstly, we need everything in $A+C$ to be less than everything in $B+D$. This is true: if $a+c\in A+C$, and $b+d\in B+D$, then since $a<b$ and $c<d$, we have $a+c<b+d$.

Next, we need $A+C$ and $B+D$ together to contain all the rationals. This is true:

Finally, we need $(A+C,B+D)$ to be one-sided: that is, $A+C$ needs to have no top element, or equivalently, if $a+c\in A+C$ then we can find a bigger $a^{\prime }+c^{\prime }$ in $A+C$. This is also true: if $a+c$ is an element of $A+C$, then we can find an element $a^{\prime }$ of $A$ which is bigger than $a$, and an element $c^{\prime }$ of $C$ which is bigger than $C$ (since both $A$ and $C$ have no top elements, because the respective Dedekind cuts are one-sided); then $a^{\prime }+c^{\prime }$ is in $A+C$ and is bigger than $a+c$.

Multiplication

Ordering

Additive commutative group structure

Ring structure

Field structure

Ordering on the field

1. ^{^︎}

   Recall: "one-sided" means that $A$ has no greatest element.