You are a person after

R L Moore's own heart.

He has been called

the greatest math teacher ever by Keith Devlin.

Moore was one of the founding fathers of

*Point Set Topology*. His foundational book has no empty point set in it. He was stead fast in his denial it was possible. Look at the list of his students. You see there some of the most important mathematicians of the 20th century. On that list is Mary Ellen Rudin who was married to your offed quoted Walter Rudin. Moore is the only person who has had five of his students to be president of the MAA.

Of course, all of mathematics is the result of the human brain. It is from definitions and/or axioms. Existence is not an issue. The empty set is very much part of modern mathematics.