elementary number theory – Why has no one mentioned the disjoint union of nothingness?

Upon talking with a colleague, I was quite blown away as originally I said this.


This is the disjoint union of two empty sets to equal the set of real numbers.

But that is obviously not true because


And that doesn’t look like the real numbers. So instead call:


Then take again the disjoint union of this set:


Then take it again with $emptyset_4$


I hope some of you know binary (backwards) because we have labelled these elements as numbers already such that each element is a number from ${0,1,2,3,4,5,6,7,8}$

We keep going and going then my first statement becomes true; therefore:


Has anyone else thought of this before? This specifically excludes the negative numbers since those aren’t natural, neither is 0 but it still is here by the equation.

Why have we not spoken about this? Why is it that our whole numbers are naturally made from disjoint unioning nothingness?