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.
A recursive definition of well-founded hereditarily finite sets is as follows:
and only sets that can be built by a finite number of applications of these two rules are hereditarily finite.
This class of sets is naturally ranked by the number of bracket pairs necessary to represent the sets:
In this way, the number of sets with bracket pairs is
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 or 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 .
The class of all hereditarily finite sets is denoted by , meaning that the cardinality of each member is smaller than . (Analogously, the class of hereditarily countable sets is denoted by .) It can also be denoted by , which denotes the th stage of the von Neumann universe.
The class is countable.
In 1937, Wilhelm Ackermann introduced an encoding of hereditarily finite sets as natural numbers. It is defined by a function that maps each hereditarily finite set to a natural number, given by the following recursive definition:
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 is mapped to .
The inverse of , which maps natural numbers back to sets, is
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, (where is the converse relation of BIT, swapping its two arguments) models Zermelo–Fraenkel set theory without the axiom of infinity. Here, each natural number models a set, and the BIT relation models the membership relation between sets.
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 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.
In the common axiomatic set theory approaches, the empty set also represents the first von Neumann ordinal number, denoted . All finite von Neumann ordinals are indeed hereditarily finite and, thus, so is the class of sets representing the natural numbers. In other words, includes each element in the standard model of natural numbers and a set theory expressing must contain all of those.
Now note that Robinson arithmetic can already be interpreted in , the very small sub-theory of with its axioms given by Extensionality, Empty Set and Adjunction. 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, thus proving that the axiom of infinity is not a consequence of the other axioms of .
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.
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