It reminded me of something I thought years ago. I guess I got the impression that if you mix matter and anti-matter, you get nothing. Therefore you should be able to split nothing into matter and anti-matter. Isn't that fun?

You're in good company. Asimov postulated a four-lobed universe (nothing splits to become energy and anti-energy, which in turn split to become matter and anti-matter and (anti-)matter and (anti-)anti-matter) in an essay in Science, Numbers, and I, published in 1966. :-)