and since is inductive. Thus . Let denote this unique element. This definition is convenient because the principle of induction immediately follows: If is inductive, then also , so that . Both these methods produce systems which satisfy the axioms of second-order arithmetic, since the axiom of power set allows us to quantify over the power set of , as in second-order logic. Thus they both completely determine isomorphic systems, and since they are isomorphic under the identity map, they must in fact be equal. An apparently weaker version Some old texts use an apparently weaker version of the axiom of infinity, to wit: This says that there is an element in x and for every element y of x there is another element of x which is a strict superset of y. This implies that x is an infinite set without saying much about its structure. However, with the help of the other axioms of ZF, we can show that this implies the existence of ω. First, if we take the powerset of any infinite set x, then that powerset will contain elements which are subsets of x of every finite cardinality (among other subsets of x). Proving the existence of those finite subsets may require either the axiom of separation or the axioms of pairing and union. Then we can apply the axiom of replacement to replace each element of that powerset of x by the initial ordinal number of the same cardinality (or zero, if there is no such ordinal). The result will be an infinite set of ordinals. Then we can apply the axiom of union to that to get an ordinal greater than or equal to ω. Independence The axiom of infinity cannot be proved from the other axioms of ZFC if they are consistent. (To see why, note that ZFC Con(ZFC – Infinity) and use Gödel's Second incompleteness theorem.) The negation of the axiom of infinity cannot be derived from the rest of the axioms of ZFC, if they are consistent. (This is tantamount to saying that ZFC is consistent, if the other axioms are consistent.) We believe this, but cannot prove it (if it is true). Indeed, using the von Neumann universe, we can build a model of ZFC – Infinity + (¬Infinity). It is , the class of hereditarily finite sets, with the inherited membership relation. Note that if the axiom of