Hereditarily finite set

In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to the empty set.

Formal definition

A recursive definition of well-founded hereditarily finite sets is as follows:

Base case: The empty set is a hereditarily finite set.

Recursion rule: If

a_{1},\dots a_{k}

are hereditarily finite, then so is

\{a_{1},\dots a_{k}\}

.

Only sets that can be built by a finite number of applications of these two rules are hereditarily finite.

Representation

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

$\{\}$ (i.e. $\emptyset$ , the Neumann ordinal "0")
$\{\{\}\}$ (i.e. $\{\emptyset \}$ or $\{0\}$ , the Neumann ordinal "1")
$\{\{\{\}\}\}$
$\{\{\{\{\}\}\}\}$ and then also $\{\{\},\{\{\}\}\}$ (i.e. $\{0,1\}$ , the Neumann ordinal "2"),
$\{\{\{\{\{\}\}\}\}\}$ , $\{\{\{\},\{\{\}\}\}\}$ as well as $\{\{\},\{\{\{\}\}\}\}$ ,
... sets represented with $6$ bracket pairs, e.g. $\{\{\{\{\{\{\}\}\}\}\}\}$ . There are six such sets
... sets represented with $7$ bracket pairs, e.g. $\{\{\{\{\{\{\{\}\}\}\}\}\}\}$ . There are twelve such sets
... sets represented with $8$ bracket pairs, e.g. $\{\{\{\{\{\{\{\{\}\}\}\}\}\}\}\}$ or $\{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}$ (i.e. $\{0,1,2\}$ , the Neumann ordinal "3")
... etc.

In this way, the number of sets with $n$ bracket pairs is^[1]

1, 1, 1, 2, 3, 6, 12, 25, 52, 113, 247, 548, 1226, 2770, 6299, 14426, ...

Discussion

The set $\{\{\},\{\{\{\}\}\}\}$ is an example for such a hereditarily finite set and so is the empty set $\{\}$ , as noted. On the other hand, the sets $\{7,{\mathbb {N} },\pi \}$ or $\{3,\{{\mathbb {N} }\}\}$ are examples of finite sets that are not hereditarily finite. For example, the first cannot be hereditarily finite since it contains at least one infinite set as an element, when ${\mathbb {N} }=\{0,1,2,\dots \}$ .

The class of all hereditarily finite sets is denoted by $H_{\aleph _{0}}$ , meaning that the cardinality of each member is smaller than $\aleph _{0}$ . (Analogously, the class of hereditarily countable sets is denoted by $H_{\aleph _{1}}$ .) $H_{\aleph _{0}}$ is in bijective correspondence with $\aleph _{0}$ . It can also be denoted by $V_{\omega }$ , which denotes the $\omega$ th stage of the von Neumann universe.^[2] So here it is a countable set.

Models

Ackermann coding

In 1937, Wilhelm Ackermann introduced an encoding of hereditarily finite sets as natural numbers.^[3]^[4]^[5] It is defined by a function $f\colon H_{\aleph _{0}}\to \omega$ that maps each hereditarily finite set to a natural number, given by the following recursive definition:

\displaystyle f(a)=\sum _{b\in a}2^{f(b)}

For example, the empty set $\{\}$ contains no members, and is therefore mapped to an empty sum, that is, the number zero. On the other hand, a set with distinct members $a,b,c,\dots$ is mapped to $2^{f(a)}+2^{f(b)}+2^{f(c)}+\ldots$ .

The inverse is given by

\displaystyle f^{-1}\colon \omega \to H_{\aleph _{0}}

\displaystyle f^{-1}(i)=\{f^{-1}(j)\mid {\text{BIT}}(i,j)=1\}

where BIT denotes the BIT predicate.

The Ackermann coding can be used to construct a model of finitary set theory in the natural numbers. More precisely, $(\mathbb {N} ,{\text{BIT}}^{\top })$ (where ${\text{BIT}}^{\top }$ is the converse relation of ${\text{BIT}}$ , swapping its two arguments) models Zermelo–Fraenkel set theory ZF without the axiom of infinity. Here, each natural number models a set, and the ${\text{BIT}}$ relation models the membership relation between sets.

Graph models

The class $H_{\aleph _{0}}$ 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 $\{\dots \}$ 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. $\{t,t,s\}=\{t,s\}$ , trivializing the permutation of the two subgraphs of shape $t$ ). This graph model enables an implementation of ZF without infinity as data types and thus an interpretation of set theory in expressive type theories.

Graph models exist for ZF and also set theories different from Zermelo set theory, such as 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.

Axiomatizations

Theories of finite sets

In the common axiomatic set theory approaches, the empty set $\{\}$ also represents the first von Neumann ordinal number, denoted $0$ . All finite von Neumann ordinals are indeed hereditarily finite and, thus, so is the class of sets representing the natural numbers. In other words, $H_{\aleph _{0}}$ includes each element in the standard model of natural numbers and so a set theory expressing $H_{\aleph _{0}}$ must necessarily contain them as well.

Now note that Robinson arithmetic can already be interpreted in ST, the very small sub-theory of Zermelo set theory Z⁻ with its axioms given by Extensionality, Empty Set and Adjunction. All of $H_{\aleph _{0}}$ has a constructive axiomatization involving these axioms and e.g. Set induction and 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 ${\mathsf {ZF}}$ , this establishes that the axiom of infinity is not a consequence these other ${\mathsf {ZF}}$ axioms.

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 $V_{\omega }$ . Note that this is also a set in this context.

If we denote by $\wp (S)$ the power set of $S$ , and by $V_{0}$ the empty set, then $V_{\omega }$ can be obtained by setting $V_{i+1}=\wp (V_{i})$ for each integer $i\geq 0$ . Thus, $V_{\omega }$ can be expressed as

\displaystyle V_{\omega }=\bigcup _{k=0}^{\infty }V_{k}

and all its elements are finite.

This formulation shows, again, that there are only countably many hereditarily finite sets: $V_{n}$ is finite for any finite $n$ , its cardinality is $2\uparrow \uparrow (n-1)$ in Knuth's up-arrow notation (a tower of $n-1$ powers of two), 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.

References

^ Sloane, N. J. A. (ed.). "Sequence A004111". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ "hereditarily finite set". nLab. January 2023. Retrieved January 28, 2023. The set of all (well-founded) hereditarily finite sets (which is infinite, and not hereditarily finite itself) is written $V_{\omega }$ to show its place in the von Neumann hierarchy of pure sets.
^ Ackermann, Wilhelm (1937). "Die Widerspruchsfreiheit der allgemeinen Mengenlehre". Mathematische Annalen. 114: 305–315. doi:10.1007/bf01594179. S2CID 120576556. Retrieved 2012-01-09.
^ Kirby, Laurence (2009). "Finitary Set Theory". Notre Dame Journal of Formal Logic. 50 (3): 227–244. doi:10.1215/00294527-2009-009.
^ Omodeo, Eugenio G.; Policriti, Alberto; Tomescu, Alexandru I. (2017). "3.3: The Ackermann encoding of hereditarily finite sets". On Sets and Graphs: Perspectives on Logic and Combinatorics. Springer. pp. 70–71. doi:10.1007/978-3-319-54981-1. ISBN 978-3-319-54980-4. MR 3558535.

[1] Sloane, N. J. A. (ed.). "Sequence A004111". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[2] "hereditarily finite set". nLab. January 2023. Retrieved January 28, 2023. The set of all (well-founded) hereditarily finite sets (which is infinite, and not hereditarily finite itself) is written $V_{\omega }$ to show its place in the von Neumann hierarchy of pure sets.

[ackermann-3] Ackermann, Wilhelm (1937). "Die Widerspruchsfreiheit der allgemeinen Mengenlehre". Mathematische Annalen. 114: 305–315. doi:10.1007/bf01594179. S2CID 120576556. Retrieved 2012-01-09.

[4] Kirby, Laurence (2009). "Finitary Set Theory". Notre Dame Journal of Formal Logic. 50 (3): 227–244. doi:10.1215/00294527-2009-009.

[5] Omodeo, Eugenio G.; Policriti, Alberto; Tomescu, Alexandru I. (2017). "3.3: The Ackermann encoding of hereditarily finite sets". On Sets and Graphs: Perspectives on Logic and Combinatorics. Springer. pp. 70–71. doi:10.1007/978-3-319-54981-1. ISBN 978-3-319-54980-4. MR 3558535.

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