Is there an empty set, and how is a definition of âˆˆ checked for validity?

There is no empty set. Empty set is a contradiction in terms. Empty set is a definition of a set.

A herd of elephants is a set. An empty herd of elephants is a definition of a set.

An empty herd of elephants is not the same as an empty herd of cows. A herd of elephants can not contain cows.

\(\displaystyle \epsilon\) is membership, it's axiomatic. Any definition that makes sense and can be understood and satisfies a simple intelligible axiom system qualifies. The essence of \(\displaystyle \epsilon\) is that it is different than =. Halmos, Suppe, ZFC, etc don't make the distinction.

Beyond that it gets too abstract, virtually meaningless symbolism:

What is the mathematics of two different symbols which can be placed between any two things other than the two symbols? This leads to a set of rules about symbols. As you get more and more rules to cover every eventuality you have to question consistency.

On the positive side, if you can get a group of people to start coming up with and agreeing on a set of rules (axioms), you can call them the foundations of mathematics and invoke them in response to questions about mathematics, implying that anyone who doesn't "know" them doesn't know mathematics. In a way it's like abstract painting and the implication that if you don't understand abstract paintings you don't understand art.

However, I can understand that some people might enjoy the APPEARANCE of abstract art, just as some people might enjoy the APPEARANCE of arranged symbols. But I have no quarrel with such people, as long as they don't intrude on Art and Science.

It may happen that a symbolism system whose elements are given real interpretations may actually come up with a verifiable prediction. In that case it could be considered A methodology of science, but not a foundation of mathematics.

Finally, to protect against obvious criticism, my views are not absolute.

Does that explain \(\displaystyle \epsilon\)?

EDIT: Empty set may have a relation to sets like 0 to the natural numbers 1,2,3... ; 0 is not a natural number. But I have to think about it.