Transitive set

In set theory, a branch of mathematics, a set ${\displaystyle A}$ is called transitive if either of the following equivalent conditions hold:

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

Similarly, a class ${\displaystyle M}$ is transitive if every element of ${\displaystyle M}$ is a subset of ${\displaystyle M}$.

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 ${\displaystyle V_{\alpha }}$ and ${\displaystyle L_{\alpha }}$ leading to the construction of the von Neumann universe ${\displaystyle V}$ and Gödel's constructible universe ${\displaystyle L}$ are transitive sets. The universes ${\displaystyle V}$ and ${\displaystyle L}$ themselves are transitive classes.

This is a complete list of all finite transitive sets with up to 20 brackets:^[1]

• ${\displaystyle \{\},}$
• ${\displaystyle \{\{\}\},}$
• ${\displaystyle \{\{\},\{\{\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\}\},\{\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\},\{\{\}\}\}\},\{\{\},\{\{\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\}\},\{\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\},\{\{\}\}\},\{\{\},\{\{\},\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\},\{\{\}\}\},\{\{\{\}\},\{\{\},\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\},\{\{\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\},\{\{\}\}\},\{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\{\{\{\}\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\},\{\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\},\{\{\}\}\}\},\{\{\},\{\{\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\}\}\},\{\{\},\{\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\},\{\{\{\{\}\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\{\}\},\{\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\}\}\},\{\{\},\{\{\},\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\}\}\},\{\{\{\}\},\{\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\{\}\},\{\{\{\{\}\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\},\{\{\}\},\{\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\},\{\{\{\}\}\}\}\},\{\{\},\{\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\}\}\},\{\{\{\}\},\{\{\},\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\}\}\},\{\{\},\{\{\}\},\{\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\{\}\}\}\},\{\{\{\}\},\{\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\{\{\}\}\},\{\{\{\{\}\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\},\{\{\}\},\{\{\{\{\}\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\}\}\},\{\{\{\{\}\}\},\{\{\},\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\}\}\},\{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\{\}\}\}\},\{\{\},\{\{\},\{\{\{\}\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\{\}\}\}\},\{\{\},\{\{\}\},\{\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\{\},\{\{\}\}\}\}\},\{\{\{\},\{\{\}\}\}\},\{\{\},\{\{\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\},\{\{\}\}\}\},\{\{\},\{\{\}\}\},\{\{\},\{\{\},\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\},\{\{\{\}\}\},\{\{\{\{\}\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\},\{\{\{\}\}\}\}\},\{\{\{\}\},\{\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\}\}\},\{\{\},\{\{\{\}\}\},\{\{\},\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\{\}\}\}\},\{\{\{\}\},\{\{\},\{\{\{\}\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\}\},\{\{\{\}\}\}\},\{\{\},\{\{\}\},\{\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\},\{\{\}\}\}\},\{\{\},\{\{\}\}\},\{\{\},\{\{\{\},\{\{\}\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\},\{\{\}\}\}\},\{\{\},\{\{\}\}\},\{\{\{\}\},\{\{\},\{\{\}\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\{\{\{\}\}\}\}\},\{\{\},\{\{\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\{\},\{\{\}\}\}\},\{\{\},\{\{\}\}\}\},}$
• ${\displaystyle \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\{\{\}\}\}\},\{\{\},\{\{\}\}\},\{\{\},\{\{\{\}\}\}\}\}.}$

Properties

A set ${\displaystyle X}$ is transitive if and only if ${\textstyle \bigcup X\subseteq X}$, where ${\textstyle \bigcup X}$ is the union of all elements of ${\displaystyle X}$ that are sets, ${\textstyle \bigcup X=\{y\mid \exists x\in X:y\in x\}}$.

If ${\displaystyle X}$ is transitive, then ${\textstyle \bigcup X}$ is transitive.

If ${\displaystyle X}$ and ${\displaystyle Y}$ are transitive, then ${\displaystyle X\cup Y}$ and ${\displaystyle X\cup Y\cup \{X,Y\}}$ are transitive. In general, if ${\displaystyle Z}$ is a class all of whose elements are transitive sets, then ${\textstyle \bigcup Z}$ and ${\textstyle Z\cup \bigcup Z}$ are transitive. (The first sentence in this paragraph is the case of ${\displaystyle Z=\{X,Y\}}$.)

A set ${\displaystyle X}$ that does not contain urelements is transitive if and only if it is a subset of its own power set, ${\textstyle X\subseteq {\mathcal {P}}(X).}$ The power set of a transitive set without urelements is transitive.

Transitive closure

The transitive closure of a set ${\displaystyle X}$ is the smallest (with respect to inclusion) transitive set that includes ${\displaystyle X}$ (i.e. ${\textstyle X\subseteq \operatorname {TC} (X)}$).^[2] Suppose one is given a set ${\displaystyle X}$, then the transitive closure of ${\displaystyle X}$ is

$${\displaystyle \operatorname {TC} (X)=\bigcup \left\{X,\;\bigcup X,\;\bigcup \bigcup X,\;\bigcup \bigcup \bigcup X,\;\bigcup \bigcup \bigcup \bigcup X,\ldots \right\}.}$$

Proof. Denote ${\textstyle X_{0}=X}$ and ${\textstyle X_{n+1}=\bigcup X_{n}}$. Then we claim that the set

$${\displaystyle T=\operatorname {TC} (X)=\bigcup _{n=0}^{\infty }X_{n}}$$

is transitive, and whenever ${\textstyle T_{1}}$ is a transitive set including ${\textstyle X}$ then ${\textstyle T\subseteq T_{1}}$.

Assume ${\textstyle y\in x\in T}$. Then ${\textstyle x\in X_{n}}$ for some ${\textstyle n}$ and so ${\textstyle y\in \bigcup X_{n}=X_{n+1}}$. Since ${\textstyle X_{n+1}\subseteq T}$, ${\textstyle y\in T}$. Thus ${\textstyle T}$ is transitive.

Now let ${\textstyle T_{1}}$ be as above. We prove by induction that ${\textstyle X_{n}\subseteq T_{1}}$ for all ${\displaystyle n}$, thus proving that ${\textstyle T\subseteq T_{1}}$: The base case holds since ${\textstyle X_{0}=X\subseteq T_{1}}$. Now assume ${\textstyle X_{n}\subseteq T_{1}}$. Then ${\textstyle X_{n+1}=\bigcup X_{n}\subseteq \bigcup T_{1}}$. But ${\textstyle T_{1}}$ is transitive so ${\textstyle \bigcup T_{1}\subseteq T_{1}}$, hence ${\textstyle X_{n+1}\subseteq T_{1}}$. This completes the proof.

Note that this is the set of all of the objects related to ${\displaystyle X}$ 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: ${\displaystyle x}$ is a transitive closure of ${\displaystyle y}$ iff ${\displaystyle x}$ is an intersection of all transitive supersets of ${\displaystyle y}$ (that is, every transitive superset of ${\displaystyle y}$ contains ${\displaystyle x}$ 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.

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.^{[clarification needed][3]}

See also

• End extension
• Transitive relation
• Supertransitive class

References

• 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 0-486-46624-8, Zbl 0259.02051

• v
• t
• e

Set theory

Overview

• Set (mathematics)

Axioms

• Adjunction
• Choice
  • countable
  • dependent
  • global
• Constructibility (V=L)
• Determinacy
• 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 · 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