Hereditarily finite set

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In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets.

Formal definition

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

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

Recursion rule: If a₁,...,a_k are hereditarily finite, then so is {a₁,...,a_k}.

The set of all well-founded hereditarily finite sets is denoted V_ω. If we denote P(S) for the power set of S, V_ω can also be constructed by first taking the empty set written V₀, then V₁ = P(V₀), V₂ = P(V₁),..., V_k = P(V_k−1),... Then

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

Discussion

The hereditarily finite sets are a subclass of the Von Neumann universe. They are a model of the axioms consisting of the axioms of set theory with the axiom of infinity replaced by its negation, thus proving that the axiom of infinity is not a consequence of the other axioms of set theory.

Notice that there are countably many hereditarily finite sets, since V_n 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. V_ω is also symbolized by $H_{{\aleph _{0}}}$ , meaning hereditarily of cardinality less than $\aleph _{0}$ .

Ackermann's bijection

Ackermann (1937) gave the following natural bijection f from the natural numbers to the hereditarily finite sets, known as the Ackermann coding. It is defined recursively by

f(2^{a}+2^{b}+\cdots )=\{f(a),f(b),\ldots \}

if a, b, ... are distinct.

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.

Rado graph

The graph whose vertices are the hereditarily finite sets, with an edge joining two vertices whenever one is contained in the other, is the Rado graph or random graph.

References

Ackermann, Wilhelm (1937), "Die Widerspruchsfreiheit der allgemeinen Mengenlehre", Mathematische Annalen, 114 (1): 305–315, doi:10.1007/BF01594179

Set theory

Axioms	Choice countable dependent Constructibility (V=L) Determinacy Extensionality Infinity Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification

Operations	Cartesian product Complement De Morgan's laws Disjoint union Intersection Power set Set difference Symmetric difference Union

Concepts Methods	Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Transfinite induction Venn diagram

Set types	Countable Empty Finite (hereditarily) Fuzzy Infinite Recursive Subset · Superset Transitive Uncountable Universal

Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck

Paradoxes Problems	Russell's paradox Suslin's problem

Set theorists	Abraham Fraenkel Bertrand Russell Ernst Zermelo Georg Cantor John von Neumann Kurt Gödel Paul Bernays Paul Cohen Richard Dedekind Thomas Jech Thoralf Skolem Willard Quine

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