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.

$emptysetsqcupemptyset=mathbb{R}$

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

But that is obviously not true because

$emptysetsqcupemptyset={(emptyset,0),(emptyset,1)}$

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

${(emptyset,0),(emptyset,1)}=emptyset_2$

Then take again the disjoint union of this set:

$emptyset_2sqcupemptyset_2={(emptyset,0,0),(emptyset,1,0),(emptyset,0,1),(emptyset,1,1)}=emptyset_4$

Then take it again with $emptyset_4$

$emptyset_4sqcupemptyset_4={(emptyset,0,0,0),(emptyset,1,0,0),(emptyset,0,1,0),(emptyset,1,1,0),(emptyset,0,0,1),(emptyset,1,0,1),(emptyset,0,1,1),(emptyset,1,1,1)}=emptyset_8$

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:

$emptysetsqcup_inftyemptyset=mathbb{R}^+cup{0}=mathbb{W}$

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?