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Some Axioms

• Two sets are equal, iff they have the same elements:

  forall x, y: x = y <=> (forall z: z in x <=> z in y).

• There exists a set that does not have any elements:

  exists x: forall y: y not  in x.

  • We call this set empty set:

    0:= such x: forall y: y not  in x.

  • Because of second axiom, 0 is well defined:

    forall y: y not  in 0.

  • Because of first axiom, 0 is unique:

    forall z: (forall y: y not  in z) => z = 0.