Recall that a Dedekind cut is defined as a subset $A$ of $\mathbb{Q}$ satisfying the following three properties:
- $A\neq \emptyset$ and $A\neq \mathbb{Q}$
- If $x\in A$, and $y < x$, then $y\in A$
- For all $x\in A$, there exists $y\in A$ s.t $y > x$
It seems to me, using the Dedekind Cut construction of the real numbers, that the standard way of defining the Supremum Partial Function object, i.e
$\sup (A) = \bigcup A$, defined for all $A\subseteq \mathbb{R}$ with an upper bound and nonempty (a partial function from $\mathcal{P}(\mathbb{R})$ to $\mathbb{R}$), can easily be modified to a total function, and yet no author ever seems to mention this.
If $\overline{\mathbb{R}} := \mathbb{R} \cup \{\mathbb{Q}, \emptyset\}$, i.e the set of all "pseudo-Dedekind cuts" (with the condition 1 above removed), and the order is again formed from strict inclusion, then $\sup (A) = \bigcup A$ in fact is the least upper bound in $\overline{\mathbb{R}}$, and forms a total function from $\mathcal{P}(\overline{\mathbb{R}})$ to $\overline{\mathbb{R}}$.
Even for the empty set, the arbitrary union above will return.... the empty set, which is in a sense the least upper bound of nothing: the smallest element of $\overline{\mathbb{R}}$.
I have a few questions.
- Why is this not how supremum is defined (at least for $\mathbb{R}$?) It seems nicer to say that supremum is a total function from $\mathcal{P}(\overline{\mathbb{R}})$ to $\overline{\mathbb{R}}$ than to say that it's a partial function.
- If we make the very natural symbolic substitution that $\infty = $ the maximum of $\overline{\mathbb{R}}$, and $-\infty = $the minimum of $\mathbb{R}$, then is it in some sense true that $\infty = \mathbb{Q}$, and $-\infty = \emptyset$?