An axiomatic system, in fact the canonical axioms, for describing what constitutes a set. The axioms state:

1. 1. Two sets are equal if they have the same elements.

2. 2. There is a set with no members, the empty set.

3. 3. Given two sets, there is a set containing just those sets as elements.

4. 4. Given two sets, there is a set whose elements are those of one or other set.

5. 5. Given a set, there is a set whose elements are the subsets of the set.

6. 6. The image of a set under a (definable) function is a set.

7. 7. There is no infinite descending sequence for set membership.

8. 8. There is an infinite set.

(Here the language of some of these axioms is necessarily somewhat informal.) The above are the ZF axioms. They circumvent issues such as Russell’s paradox by not permitting Russell’s paradoxical ‘set’ to be a set. Both the axiom of choice and the continuum hypothesis are independent of the ZF axioms.