I am struggling to understand why the Additive Inverse is typically defined as such:
$$\alpha^∗ := \{x \in \mathbb Q | \exists r > 0\text{ such that }−x−r\notin \alpha\}$$
or in another form
$$\alpha^* := \{−p−r : p\notin \alpha, r>0\}$$
or from my book
$$−\alpha = \{r \in \mathbb Q : r < -s\text{ for some }s\notin\alpha\}$$
For clarity,
$$\begin{array} a\alpha :=& \{x\in\mathbb Q | x < a\text{ for some }a\in\mathbb R\}\\0^* :=& \{x\in\mathbb Q| x < 0\}\end{array}$$
So, $α^∗+α = 0^*$
I've dug around and found this:
motivation of additive inverse of a Dedekind cut set
which I already understood from the get-go. I know that $α^∗$ (the additive inverse) is defined as such because if not, then the complement $α^∗$ of a rational cut $α$ would contain the element $a$. Since $a$ is rational, then the complement $α^∗$ would have a greatest rational $a$. Also, to further extend this reasoning, since $0^*$ is defined for elements strictly less than $0$, if $a$ was included in the complement, then there may be two elements in $α^∗$ and $α$ such that these two elements add to $0$, which is not in $0^*$. So, I get all of that. I'm wondering, why is the additive inverse not just this?
$$α^* := \{x∈Q | x < -a \text{ for some } a∈\Bbb R\}$$
It just seems much less fussy than the above definition of $α^*$ which takes the complement of $α$, then creates a negative image of the complement, and creates a smaller subset of that. It just seems so unnecessary. My definition of $α^*$ seems to fit all of the criteria of a Dedekind cut: it's non empty, it's not $\Bbb Q$, it is closed below, and it contains no largest element. Also, when added to $α$, it gives $0^*$.
What am I missing here?