A collection of elements which can be proven to be completed by the axioms of set theory and the theorems that can be proven from them.

The collection of every object which meets a certain condition or predicate is not a set, it is a class. If you don't define things this way, you let youself in for A blather of paradoxes when infinite sets come into play. However, many classes are represented by sets. Indeed, most of set theory is dedicated to determining which (infinite) classes are represented by (infinite) sets.

Modern set theory contains no concept of an "individual element". In other words, the only things that can be elements of sets are other sets. This is quite a far cry from the set theory taught in first grade, built around collections of real objects!

The essential property of a set is elementhood. That is, a set (call it S) is described by saying something like: "This is in S. Also, this other thing is in S. Also everything in this other set is in S..."

The assertion that a certain set a is an element of another set b is symbolized

a ∈ b

or, for browsers that can't handle HTML character entities for mathematical symbols,

a e b

The definition of a set allows for no repetition of elements. A set is either a member of another set, or it is not.

Elementhood can be used to build more complex statements about sets:

a = b
"a equals b"
if every element of a is an element of b, and vice versa.
a <= b
a ⊆ b
"a is a subset of b"
if every element of a is an element of b.
a < b
a ⊂ b
"a is a proper subset of b"
if every element of a is an element of b, and a != b.
a U b
a ∪ b
"a union b"
is the set of all sets that are in a, b, or both.
a * b
a ∩ b
"a intersection b"
is the set of all sets that are in both a and b.
ab
is the set of all mappings from a subset of a to a subset of b.