Hereditarily finite set

The class ${\displaystyle H_{\aleph _{0}}}$ is countable. Ackermann (1937) gave the following natural bijection f from the natural numbers to the ${\displaystyle H_{\aleph _{0}}}$, known as the Ackermann coding. It is defined recursively by

${\displaystyle f(2^{a}+2^{b}+\cdots )=\{f(a),f(b),\ldots \}}$ if a, b, ... are distinct.

E.g.

${\displaystyle f(5)=f(4+1)=f(2^{2}+2^{0})=\{f(2),f(0)\}=\{f(2^{1}),\{\}\}=\{\{f(1)\},\{\}\}=\{\{f(2^{0})\},\{\}\}=\{\{\{f(0)\}\},\{\}\}=\{\{\{\{\}\}\},\{\}\}}$

We have f(m) ∈ f(n) if and only if the mth binary digit of n (counting from the right starting at 0) is 1.

This class of sets is naturally ranked by the number of bracket pairs necessary to represent the sets:

• ${\displaystyle \{\}}$ (i.e. ${\displaystyle \emptyset }$, i.e. the Neumann ordinal "0"),
• ${\displaystyle \{\{\}\}}$ (i.e. ${\displaystyle \{\emptyset \}}$, i.e. Neumann ordinal "1"),
• ${\displaystyle \{\{\{\}\}\}}$,
• ${\displaystyle \{\{\{\{\}\}\}\}}$ and then also ${\displaystyle \{\{\},\{\{\}\}\}}$ (i.e. Neumann ordinal "2"),
• ${\displaystyle \{\{\{\{\{\}\}\}\}\}}$, ${\displaystyle \{\{\{\},\{\{\}\}\}\}}$ as well as ${\displaystyle \{\{\},\{\{\{\}\}\}\}}$,
• ... set represented with ${\displaystyle 6}$ bracket pairs,
• ... set represented with ${\displaystyle 7}$ bracket pairs, e.g. ${\displaystyle \{\{\{\{\{\{\{\}\}\}\}\}\}\}}$ or ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\}\}}$ (i.e. Neumann ordinal "3"),
• ... etc.

In this way, the number of sets with ${\displaystyle n}$ bracket pairs is[1] ${\displaystyle 1,1,1,2,3,6,12,25,52,113,247,548,1226,2770,6299,14426,\dots }$

The set ${\displaystyle \emptyset }$ also represents the first von Neumann ordinal number, denoted ${\displaystyle 0}$. And indeed all finite von Neumann ordinals are in ${\displaystyle H_{\aleph _{0}}}$ 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 ${\displaystyle Z^{-}}$ with axioms given by Extensionality, Empty Set and Adjunction.

Indeed, ${\displaystyle H_{\aleph _{0}}}$ 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.

${\displaystyle ~V_{4}~}$ represented with circles in place of curly brackets    

The hereditarily finite sets are a subclass of the Von Neumann universe. Here, all well-founded hereditarily finite sets is denoted V_ω. Note that this is a set in this context.

If we denote by ℘(S) the power set of S, and by V₀ the empty set, then V_ω can be obtained by setting V₁ = ℘(V₀), V₂ = ℘(V₁),..., V_k = ℘(V_k−1),... and so on.

Thus, V_ω can be expressed as ${\displaystyle V_{\omega }=\bigcup _{k=0}^{\infty }V_{k}}$.

We see, again, that there's only countably many hereditarily finite sets: V_n is finite for any finite n. Its cardinality is ⁿ⁻¹2, see tetration. The union of countably many finite sets is countable.

Equivalently, a set is hereditarily finite if and only if its transitive closure is finite.