h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

The most important property that nothing possesses is uniqueness.

Proof that 'nothing' is unique

This is an informal proof. It can easily be made formal by translating it into axiomatic set theory.
In this (informal) form it requires no maths knowledge and is understandable by anyone, I hope.

Proof by Contradiction (reductio ad absurdum):

1. Assume 'nothing' is not unique

2. Thus there is (at least) two distinct nothings.

3. Lets call them 'nothing1' and 'nothing2'

4. So 'nothing1' is not equal to (not identical to, not the same as) 'nothing2'

5. So 'nothing1' must have something that 'nothing2' does not or 'nothing2' must have something that 'nothing1' does not

6. Or in other words: 'nothing1' has something and 'nothing2' doesn't or 'nothing2' has something and 'nothing1' doesn't

7. invoking a law of logic which allows us to drop one of the conjuncts in a conjunction - the 'and' part):
'nothing1' has something or 'nothing2' has something

8. But that's absurd because by definition both 'nothing1' and 'nothing2' have nothing, they can't have something!

9. Thus the original assumption was false

10. Hence 'nothing' is unique.

This property of 'nothing' has very profound consequences, which pervade the whole universe.

See Jim a-Khalili's BBC documentary on 'Everything and Nothing', part2: 'Nothing'.