prove the existence of the set {Z0, Z1, Z2, ...} where Z0 is the set of natural numbers, and Zn+1 is the power set of Zn. They also introduced the axiom of replacement, which guarantees the existence of this set. However, adding axioms as they are needed neither guarantees the existence of all reasonable sets nor clarifies the difference between sets that are safe to use and collections that lead to contradictions. In a 1923 letter to Zermelo, von Neumann outlined an approach to set theory that identifies sets that are "too big" and might lead to contradictions. Von Neumann identified these sets using the criterion: "A set is 'too big' if and only if it is equivalent with the set of all things." He then restricted how these sets may be used: "... in order to avoid the paradoxes those [sets] which are 'too big' are declared to be impermissible as elements." By combining this restriction with his criterion, von Neumann obtained his first version of the axiom of limitation of size, which in the language of classes states: A class is a proper class if and only if it is equinumerous with V. By 1925, Von Neumann modified his axiom by changing "it is equinumerous with V" to "it can be mapped onto V", which produces the axiom of limitation of size. This modification allowed von Neumann to give a simple proof of the axiom of replacement. Von Neumann's axiom identifies sets as classes that cannot be mapped onto V. Von Neumann realized that, even with this axiom, his set theory does not fully characterize sets. Gödel found von Neumann's axiom to be "of great interest": "In particular I believe that his [von Neumann's] necessary and sufficient condition which a property must satisfy, in order to define a set, is of great interest, because it clarifies the relationship of axiomatic set theory to the paradoxes. That this condition really gets at the essence of things is seen from the fact that it implies the axiom of choice, which formerly stood quite apart from other existential principles. The inferences, bordering on the paradoxes, which are made possible by this way of looking at things, seem to me, not only very elegant, but also very interesting from the logical point of view. Moreover I believe that only by going farther in this direction, i.e., in the direction opposite to constructivism,