11.2.3 Dedekind reals are Dedekind complete

We obtained $\mathbb{R}_{\mathsf{d}}$ as the type of Dedekind cuts  on $\mathbb{Q}$. But we could have instead started with any archimedean   ordered field $F$ and constructed Dedekind cuts on $F$. These would again form an archimedean ordered field $\bar{F}$, the Dedekind completion of $F$, with $F$ contained as a subfield  . What happens if we apply this construction to $\mathbb{R}_{\mathsf{d}}$, do we get even more real numbers? The answer is negative. In fact, we shall prove a stronger result: $\mathbb{R}_{\mathsf{d}}$ is final.

Say that an ordered field $F$ is admissible for $\Omega$ when the strict order $<$ on $F$ is a map ${<}:F\to F\to\Omega$.

Theorem 11.2.1.

Every archimedean ordered field which is admissible for $\Omega$ is a subfield of $\mathbb{R}_{\mathsf{d}}$.

Proof.

Let $F$ be an archimedean ordered field. For every $x:F$ define $L,U:\mathbb{Q}\to\Omega$ by

 $L_{x}(q):\!\!\equiv(q

(We have just used the assumption  that $F$ is admissible for $\Omega$.) Then $(L_{x},U_{x})$ is a Dedekind cut. Indeed, the cuts are inhabited and rounded because $F$ is archimedean and $<$ is transitive     , disjoint because $<$ is irreflexive  , and located because $<$ is a weak linear order. Let $e:F\to\mathbb{R}_{\mathsf{d}}$ be the map $e(x):\!\!\equiv(L_{x},U_{x})$.

We claim that $e$ is a field embedding  which preserves and reflects the order. First of all, notice that $e(q)=q$ for a rational number $q$. Next we have the equivalences, for all $x,y:F$,

 $x

so $e$ indeed preserves and reflects the order. That $e(x+y)=e(x)+e(y)$ holds because, for all $q:\mathbb{Q}$,

 $q

The implication  from right to left is obvious. For the other direction, if $q then there merely exists $r:\mathbb{Q}$ such that $q-y, and by taking $s:\!\!\equiv q-r$ we get the desired $r$ and $s$. We leave preservation of multiplication  by $e$ as an exercise. ∎

To establish that the Dedekind cuts on $\mathbb{R}_{\mathsf{d}}$ do not give us anything new, we need just one more lemma.

Lemma 11.2.2.

If $F$ is admissible for $\Omega$ then so is its Dedekind completion.

Proof.

Let $\bar{F}$ be the Dedekind completion of $F$. The strict order on $\bar{F}$ is defined by

 $((L,U)<(L^{\prime},U^{\prime})):\!\!\equiv\exists(q:\mathbb{Q}).\,U(q)\land L^% {\prime}(q).$

Since $U(q)$ and $L^{\prime}(q)$ are elements of $\Omega$, the lemma holds as long as $\Omega$ is closed under conjunctions  and countable  existentials, which we assumed from the outset. ∎

Corollary 11.2.3.

The Dedekind reals are Dedekind complete: for every real-valued Dedekind cut $(L,U)$ there is a unique $x:\mathbb{R}_{\mathsf{d}}$ such that $L(y)=(y and $U(y)=(x for all $y:\mathbb{R}_{\mathsf{d}}$.

Proof.

By \autoreflem:cuts-preserve-admissibility the Dedekind completion $\bar{\mathbb{R}}_{\mathsf{d}}$ of $\mathbb{R}_{\mathsf{d}}$ is admissible for $\Omega$, so by \autorefRD-final-field we have an embedding $\bar{\mathbb{R}}_{\mathsf{d}}\to\mathbb{R}_{\mathsf{d}}$, as well as an embedding $\mathbb{R}_{\mathsf{d}}\to\bar{\mathbb{R}}_{\mathsf{d}}$. But these embeddings must be isomorphisms    , because their compositions are order-preserving field homomorphisms which fix the dense subfield $\mathbb{Q}$, which means that they are the identity  . The corollary now follows immediately from the fact that $\bar{\mathbb{R}}_{\mathsf{d}}\to\mathbb{R}_{\mathsf{d}}$ is an isomorphism. ∎

 Title 11.2.3 Dedekind reals are Dedekind complete \metatable