The disjoint union function is strictly defined as a binary operator that unions two disjoint sets with the strict expectation of both sets not equaling one another (well they are disjoint, so obviously).
As a formal definition from Wolfram Alpha:
"The disjoint union of two sets A and B, [$A\neq B$], is a binary operator that combines all distinct elements of a pair of given sets..."
"$A\sqcup B= (A\times\{0\})\cup(B\times\{1\})$"
My question is why don't we allow both sets be equal for number theory?
Take for example, $\mathbb{R}\sqcup\mathbb{R}$.
Let's assume we can compute this when both sets are equal to one another.
We then can conclude this: $\mathbb{R}\sqcup\mathbb{R}=\{(-\infty,0),...,(-1,0),(0,0),(1,0),...,(\infty,0),(-\infty,1),...,(-1,1),(0,1),(1,1),...,(\infty,1)\}$
From this we redefine $\mathbb{R}=\{(x,0):x\in\mathbb{R}\}$ and $\mathbb{I}=\{(y,1):y\in\mathbb{R}\}$.
Since unioned, we then can further represent it as this:
$\mathbb{R}\sqcup\mathbb{R}=\mathbb{R}\cup\mathbb{I}=\mathbb{C}$.
For if we continued we would then equate further numbers such as such as quaternions, octonions, sedenions, etc.
For if it works, then why do we limit ourselves with such restraint $A\neq B$? Why do we purposely stop ourselves from limitless potential?