e.g. . There are twelve such sets ... sets represented with bracket pairs, e.g. or (i.e. , the Neumann ordinal "3") ... etc. In this way, the number of sets with bracket pairs is Axiomatizations Theories of finite sets The set also represents the first von Neumann ordinal number, denoted . And indeed all finite von Neumann ordinals are in and thus the class of sets representing the natural numbers, i.e it includes each element in the standard model of natural numbers. Robinson arithmetic can already be interpreted in ST, the very small sub-theory of with axioms given by Extensionality, Empty Set and Adjunction. Indeed, has a constructive axiomatizations involving these axiom and e.g. Set induction and Replacement. Their models then also fulfill the axioms consisting of the axioms of Zermelo–Fraenkel set theory without the axiom of infinity. In this context, the negation of the axiom of infinity may be added, thus proving that the axiom of infinity is not a consequence of the other axioms of set theory. ZF The hereditarily finite sets are a subclass of the Von Neumann universe. Here, the class of all well-founded hereditarily finite sets is denoted Vω. Note that this is also a set in this context. If we denote by ℘(S) the power set of S, and by V0 the empty set, then Vω can be obtained by setting V1 = ℘(V0), V2 = ℘(V1),..., Vk = ℘(Vk−1),... and so on. Thus, Vω can be expressed as and all its elements are finite. We see, again, that there are only countably many hereditarily finite sets: Vn is finite for any finite n, its cardinality is n−12 (see tetration), and the union of countably many finite sets is countable. Equivalently, a set is hereditarily finite if and only if its transitive closure is finite. Graph models The class can be seen to be in exact correspondence with a class of rooted trees, namely those without non-trivial symmetries (i.e. the only automorphism is the identity): The root vertex corresponds to the top level bracket and each edge leads to an element (another such set) that can act as a root vertex in its own right. No automorphism of this graph exist, corresponding to the fact that equal branches are identified (e.g. , trivializing the permutation of the two subgraphs of shape ). This graph model enables an implementation of ZF without infinity as data