Transitive set

In set theory, a set A is transitive, if and only if

whenever x ∈ A, and y ∈ x, then y ∈ A, or, equivalently,
whenever x ∈ 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).

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 L and V themselves are transitive classes.

Properties

A set X is transitive if and only if $\bigcup X \subseteq X$ , where $\bigcup X$ is the union of all elements of X that are sets, $\bigcup X = \{y \mid (\exists x \in X) y \in x\}$ . If X is transitive, then $\bigcup X$ is transitive. If X and Y are transitive, then X∪Y∪{X,Y} is transitive. In general, if X is a class all of whose elements are transitive sets, then $X \cup \bigcup X$ is transitive.

A set X which does not contain urelements is transitive if and only if it is a subset of its own power set, $X \subset \mathcal{P}(X).$ 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 which contains X. Suppose one is given a set X, then the transitive closure of X is

\bigcup \{ X, \bigcup X, \bigcup \bigcup X, \bigcup \bigcup \bigcup X, \bigcup \bigcup \bigcup \bigcup X, \ldots \}.

Note that this is the set of all of the objects related to X by the transitive closure of the membership relation.

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. 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.^[1]

References

↑ Goldblatt (1998) p.161

Ciesielski, Krzysztof (1997), Set theory for the working mathematician, London Mathematical Society Student Texts 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 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

External links

Weisstein, Eric W., "Transitive", MathWorld.
Weisstein, Eric W., "Transitive Closure", MathWorld.
Weisstein, Eric W., "Transitive Reduction", MathWorld.

Set theory

Axioms	Choice countable dependent 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 Lotfi A. Zadeh Paul Bernays Paul Cohen Richard Dedekind Thomas Jech Willard Quine

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