Set Theory

I have a question about set theory. Suppose we are working with the
Zermelo-Frankel axioms, but we exclude the axiom of regularity. Thus,
it is conceivable that all sorts of 'crazy' sets exist. Is this proof
valid:

Suppose there is a set S such that for all x, x is an element of S. By
the axiom of serparation, let T be the set of elements 'a' in S such
that a is not an element of a. Now, we have two cases, namely T is an
element of T, and T is not an element of T. Either way, we get a
contradiction to our construction of T. So, our initial assumption
must have been false, and there is no set of all sets.

Can someone tell me "why," from a philosophical standpoint, the axiom
of regularity is included in ZF? What I mean is, can you tell me a
statement which is 'intuitively" true, but requires the axiom of
regularity? Perhaps my intuition is just off. Who here thinks that
the axiom of regularity is intuitive. What problems would we run into
if we dropped it from set theory? I realize, there could exist things
like sets which contain themselves, but what problems would this cause
in mathematics?

.

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