Transitive set

In set theory, a branch of mathematics, a set $A$ is called transitive if either of the following equivalent conditions holds:

whenever $x \in A$ , and $y \in x$ , then $y \in A$ .
whenever $x \in A$ , and $x$ is not an urelement, then $x$ is a subset of $A$ .

Similarly, a class $M$ is transitive if every element of $M$ is a subset of $M$ .

Examples

Using the definition of ordinal numbers suggested by John von Neumann, ordinal numbers are defined as hereditarily transitive sets: an ordinal number is a transitive set whose members are also transitive (and thus ordinals). The class of all ordinals is a transitive class. Any of the stages $V_{α}$ and $L_{α}$ leading to the construction of the von Neumann universe $V$ and Gödel's constructible universe $L$ are transitive sets. The universes $V$ and $L$ themselves are transitive classes. This is a complete list of all finite transitive sets with up to 20 brackets:^[1]

${},$
${{}},$
${{}, {{}}},$
${{}, {{}}, {{{}}}},$
${{}, {{}}, {{}, {{}}}},$
${{}, {{}}, {{{}}}, {{{{}}}}},$
${{}, {{}}, {{{}}}, {{}, {{}}}},$
${{}, {{}}, {{{}}}, {{}, {{{}}}}},$
${{}, {{}}, {{{}}}, {{{}}, {{{}}}}},$
${{}, {{}}, {{{}, {{}}}}, {{}, {{}}}},$
${{}, {{}}, {{{}}}, {{}, {{}}, {{{}}}}},$
${{}, {{}}, {{}, {{}}}, {{}, {{}, {{}}}}},$
${{}, {{}}, {{}, {{}}}, {{{}}, {{}, {{}}}}},$
${{}, {{}}, {{{}}}, {{{{}}}}, {{}, {{}}}},$
${{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}},$
${{}, {{}}, {{{}}}, {{{{}}}}, {{{{{}}}}}},$
${{}, {{}}, {{{}}}, {{{{}}}}, {{}, {{{}}}}},$
${{}, {{}}, {{{}}}, {{{}, {{}}}}, {{}, {{}}}},$
${{}, {{}}, {{{}}}, {{}, {{}}}, {{}, {{{}}}}},$
${{}, {{}}, {{{}}}, {{{{}}}}, {{}, {{{{}}}}}},$
${{}, {{}}, {{{}}}, {{{{}}}}, {{{}}, {{{}}}}},$
${{}, {{}}, {{{}}}, {{}, {{}}}, {{}, {{}, {{}}}}},$
${{}, {{}}, {{{}}}, {{}, {{}}}, {{{}}, {{{}}}}},$
${{}, {{}}, {{{}}}, {{{{}}}}, {{{}}, {{{{}}}}}},$
${{}, {{}}, {{{}}}, {{{{}}}}, {{}, {{}}, {{{}}}}},$
${{}, {{}}, {{{}}}, {{{}, {{{}}}}}, {{}, {{{}}}}},$
${{}, {{}}, {{{}}}, {{}, {{}}}, {{{}}, {{}, {{}}}}},$
${{}, {{}}, {{{}}}, {{}, {{}}}, {{}, {{}}, {{{}}}}},$
${{}, {{}}, {{{}}}, {{}, {{{}}}}, {{{}}, {{{}}}}},$
${{}, {{}}, {{{}}}, {{{{}}}}, {{{{}}}, {{{{}}}}}},$
${{}, {{}}, {{{}}}, {{{{}}}}, {{}, {{}}, {{{{}}}}}},$
${{}, {{}}, {{{}}}, {{}, {{}}}, {{{{}}}, {{}, {{}}}}},$
${{}, {{}}, {{{}}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}},$
${{}, {{}}, {{{}}}, {{}, {{{}}}}, {{}, {{}, {{{}}}}}},$
${{}, {{}}, {{{}}}, {{}, {{{}}}}, {{}, {{}}, {{{}}}}},$
${{}, {{}}, {{{{}, {{}}}}}, {{{}, {{}}}}, {{}, {{}}}},$
${{}, {{}}, {{{}, {{}}}}, {{}, {{}}}, {{}, {{}, {{}}}}},$
${{}, {{}}, {{{}}}, {{{{}}}}, {{}, {{{}}}, {{{{}}}}}},$
${{}, {{}}, {{{}}}, {{{{}}, {{{}}}}}, {{{}}, {{{}}}}},$
${{}, {{}}, {{{}}}, {{}, {{}}}, {{}, {{{}}}, {{}, {{}}}}},$
${{}, {{}}, {{{}}}, {{}, {{{}}}}, {{{}}, {{}, {{{}}}}}},$
${{}, {{}}, {{{}}}, {{{}}, {{{}}}}, {{}, {{}}, {{{}}}}},$
${{}, {{}}, {{{}, {{}}}}, {{}, {{}}}, {{}, {{{}, {{}}}}}},$
${{}, {{}}, {{{}, {{}}}}, {{}, {{}}}, {{{}}, {{}, {{}}}}},$
${{}, {{}}, {{{}}}, {{{{}}}}, {{{{{}}}}}, {{}, {{}}}},$
${{}, {{}}, {{{}}}, {{{{}}}}, {{{}, {{}}}}, {{}, {{}}}},$
${{}, {{}}, {{{}}}, {{{{}}}}, {{}, {{}}}, {{}, {{{}}}}} .$

Properties

A set $X$ is transitive if and only if $⋃ X \subseteq X$ , where $⋃ X$ is the union of all elements of $X$ that are sets, $⋃ X = {y ∣ \exists x \in X : y \in x}$ . If $X$ is transitive, then $⋃ X$ is transitive. If $X$ and $Y$ are transitive, then $X \cup Y$ and $X \cup Y \cup {X, Y}$ are transitive. In general, if $Z$ is a class all of whose elements are transitive sets, then $⋃ Z$ and $Z \cup ⋃ Z$ are transitive. (The first sentence in this paragraph is the case of $Z = {X, Y}$ .) A set $X$ that does not contain urelements is transitive if and only if it is a subset of its own power set, $X \subseteq 𝒫 (X) .$ The power set of a transitive set without urelements is transitive.

Transitive closure

The transitive closure of a set $X$ is the smallest (with respect to inclusion) transitive set that includes $X$ (i.e. $X \subseteq TC (X)$ ).^[2] Suppose one is given a set $X$ , then the transitive closure of $X$ is $TC (X) = ⋃ {X, ⋃ X, ⋃ ⋃ X, ⋃ ⋃ ⋃ X, ⋃ ⋃ ⋃ ⋃ X, \dots} .$ Proof. Denote $X_{0} = X$ and $X_{n + 1} = ⋃ X_{n}$ . Then we claim that the set $T = TC (X) = ⋃_{n = 0}^{\infty} X_{n}$ is transitive, and whenever $T_{1}$ is a transitive set including $X$ then $T \subseteq T_{1}$ . Assume $y \in x \in T$ . Then $x \in X_{n}$ for some $n$ and so $y \in ⋃ X_{n} = X_{n + 1}$ . Since $X_{n + 1} \subseteq T$ , $y \in T$ . Thus $T$ is transitive. Now let $T_{1}$ be as above. We prove by induction that $X_{n} \subseteq T_{1}$ for all $n$ , thus proving that $T \subseteq T_{1}$ : The base case holds since $X_{0} = X \subseteq T_{1}$ . Now assume $X_{n} \subseteq T_{1}$ . Then $X_{n + 1} = ⋃ X_{n} \subseteq ⋃ T_{1}$ . But $T_{1}$ is transitive so $⋃ T_{1} \subseteq T_{1}$ , hence $X_{n + 1} \subseteq T_{1}$ . This completes the proof. Note that this is the set of all of the objects related to $X$ by the transitive closure of the membership relation, since the union of a set can be expressed in terms of the relative product of the membership relation with itself. The transitive closure of a set can be expressed by a first-order formula: $x$ is a transitive closure of $y$ iff $x$ is an intersection of all transitive supersets of $y$ (that is, every transitive superset of $y$ contains $x$ as a subset).

Transitive models of set theory

Transitive classes are often used for construction of interpretations of set theory in itself, usually called inner models. The reason is that properties defined by bounded formulas are absolute for transitive classes.^[3] A transitive set (or class) that is a model of a formal system of set theory is called a transitive model of the system (provided that the element relation of the model is the restriction of the true element relation to the universe of the model). Transitivity is an important factor in determining the absoluteness of formulas. In the superstructure approach to non-standard analysis, the non-standard universes satisfy strong transitivity. Here, a class $𝒞$ is defined to be strongly transitive if, for each set $S \in 𝒞$ , there exists a transitive superset $T$ with $S \subseteq T \subseteq 𝒞$ . A strongly transitive class is automatically transitive. This strengthened transitivity assumption allows one to conclude, for instance, that $𝒞$ contains the domain of every binary relation in $𝒞$ .^[4]

References

↑ "Number of rooted identity trees with n nodes (rooted trees whose automorphism group is the identity group).", OEIS
↑ Ciesielski, Krzysztof (1997), Set theory for the working mathematician, Cambridge: Cambridge University Press, p. 164, ISBN 978-1-139-17313-1, OCLC 817922080
↑ Viale, Matteo (November 2003), "The cumulative hierarchy and the constructible universe of ZFA", Mathematical Logic Quarterly, 50 (1), Wiley: 99–103, doi:10.1002/malq.200310080
↑ Goldblatt (1998) p.161

Ciesielski, Krzysztof (1997), Set theory for the working mathematician, London Mathematical Society Student Texts, vol. 39, Cambridge: Cambridge University Press, ISBN 0-521-59441-3, Zbl 0938.03067
Goldblatt, Robert (1998), Lectures on the hyperreals. An introduction to nonstandard analysis, Graduate Texts in Mathematics, vol. 188, New York, NY: Springer-Verlag, ISBN 0-387-98464-X, Zbl 0911.03032
Jech, Thomas (2008) [originally published in 1973], The Axiom of Choice, Dover Publications, ISBN 978-0-486-46624-8, Zbl 0259.02051

[1] "Number of rooted identity trees with n nodes (rooted trees whose automorphism group is the identity group).", OEIS

[2] Ciesielski, Krzysztof (1997), Set theory for the working mathematician, Cambridge: Cambridge University Press, p. 164, ISBN 978-1-139-17313-1, OCLC 817922080

[3] Viale, Matteo (November 2003), "The cumulative hierarchy and the constructible universe of ZFA", Mathematical Logic Quarterly, 50 (1), Wiley: 99–103, doi:10.1002/malq.200310080

[4] Goldblatt (1998) p.161

[1]

[2]

[3]

[4]

v t e Set theory
Overview	Set (mathematics)	Venn diagram of set intersection
Axioms	Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification
Operations	Cartesian product Complement (i.e. set difference) De Morgan's laws Disjoint union Identities Intersection Power set Symmetric difference Union
Concepts Methods	Almost Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram
Set types	Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset⧼dot-separator⧽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 Burali-Forti paradox
Set theorists	Paul Bernays Georg Cantor Paul Cohen Richard Dedekind Abraham Fraenkel Kurt Gödel Thomas Jech John von Neumann Willard Quine Bertrand Russell Thoralf Skolem Ernst Zermelo

Transitive set

Contents

Examples

Properties

Transitive closure

Transitive models of set theory

See also

References

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