Editing Hereditarily finite set (section)

===Theories of finite sets===
In the common axiomatic set theory approaches, the empty set <math>\{\}</math> also represents the first von Neumann [[ordinal number]], denoted <math>0</math>. All finite von Neumann ordinals are indeed hereditarily finite and, thus, so is the class of sets representing the natural numbers. In other words, <math>H_{\aleph_0}</math> includes each element in the [[Set-theoretic definition of natural numbers|standard model of natural numbers]] and so a set theory expressing <math>H_{\aleph_0}</math> must necessarily contain them as well.

Now note that [[Robinson arithmetic]] can already be interpreted in [[General set theory|ST]], the very small sub-theory of [[Zermelo set theory]] Z<sup>&minus;</sup> with its [[axioms]] given by [[Axiom of Extensionality|Extensionality]], Empty Set and [[General set theory|Adjunction]]. All of <math>H_{\aleph_0}</math> has a [[Constructive set theory|constructive axiomatization]] involving these axioms and e.g. [[Epsilon induction|Set induction]] and [[Axiom of replacement|Replacement]].

Axiomatically characterizing the theory of hereditarily finite sets, the negation of the [[axiom of infinity]] may be added. As the theory validates the other axioms of <math>\mathsf{ZF}</math>, this establishes that the axiom of infinity is not a consequence of these other <math>\mathsf{ZF}</math> axioms.