# 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?