Definition: A set is said to be pure if no individuals belong to its transitive closure.
Well there are always individuals, right?
Consider now the following axiom candidate. Axiom of purity: Every set is pure
The main reason for the [axiom of purity] is that, as was discovered fairly early, it is is not necessary to assume the existence of individuals in order that set theory should act as a foundation of mathematics, while if we rule them out from the outset, we can simplify the theory.
A TOTAL BAN ON INDIVIDUALS
After 1963, not even set theorists had any use for individuals. Worse, there are proofs in set theory that do not work if we have to allow for them. So it is unsurprising that in the last 40 years individual have largely disappeared from view.
FULL PURITY FOR MATHEMATICS. NO INDIVIDUALS. ALL ARE SETS. THIS WILL ENSURE THAT EVERYTHING IS WELL-FOUNDED AND REGULATED.
However, we shall not follow this trend here.
The reason is that to do so would cut our theory off from at least one of its intended applications. It is by no means obvious what justifies the applicability of mathematics in general to what lies outside it…the most natural, if not only way, way to ensure that that calculus is available to be applied to counting non-mathematical things — chairs, electrons, thoughts, angels — is to allow such things into the theory as individuals.
Phew. From Set Theory and Its Philosophy.