A natural set never contains itself. The existence or non-existence of a natural set is decided using a modified "Axiom of Comprehension". This modified axiom keeps everything that contains itself out of set formation. This also includes the property of not containing itself, although every natural set has this property. If it were possible to form a set with this property, then the antinomy "contains itself" and "does not contain itself", named after Bertrand Russell, would apply to this set. While this is paradoxical, it is still a corollary when trying to form a set with the property "does not contain itself". The modified "Axiom of Comprehension" takes this fact into account and decides on the non-existence of a set to be formed with the property "does not contain itself" because it would have the property "contains itself". There can therefore be no antinomy like that of Bertrand Russell in the case of natural sets. The modified "Axiom of Comprehension" states that a natural set does not contain itself. This means that the set to be formed cannot have the property or condition required for the set to be formed. This, together with the elimination of Russell's antinomy, provides an existence criterion for natural sets. With this, essential statements about the natural sets can be proved.