Editing Hereditarily finite set (section)

===Graph models===
The class <math>H_{\aleph_0}</math> can be seen to be in exact correspondence with a class of [[Tree (graph theory)#Rooted tree|rooted trees]], namely those without non-trivial symmetries (i.e. the only [[Graph automorphism|automorphism]] is the identity): 
The root vertex corresponds to the top level bracket <math>\{\dots\}</math> and each [[Vertex (graph theory)|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. <math>\{t,t,s\}=\{t,s\}</math>, trivializing the permutation of the two subgraphs of shape <math>t</math>).
This graph model enables an implementation of ZF without infinity as data types and thus an interpretation of set theory in expressive [[type theory|type theories]].

Graph [[model theory|model]]s exist for ZF and also set theories different from Zermelo set theory, such as [[Aczel's anti-foundation axiom|non-well founded]] theories. Such models have more intricate edge structure.

In [[graph theory]], the graph whose vertices correspond to hereditarily finite sets and edges correspond to set membership is the [[Rado graph]] or random graph.