## You are here

Homecriterion for a set to be transitive

## Primary tabs

# criterion for a set to be transitive

###### Theorem.

A set $X$ is transitive if and only if its power set $\mathcal{P}(X)$ is transitive.

###### Proof.

First assume $X$ is transitive. Let $A\in B\in\mathcal{P}(X)$. Since $B\in\mathcal{P}(X)$, $B\subseteq X$. Thus, $A\in X$. Since $X$ is transitive, $A\subseteq X$. Hence, $A\in\mathcal{P}(X)$. It follows that $\mathcal{P}(X)$ is transitive.

Conversely, assume $\mathcal{P}(X)$ is transitive. Let $a\in X$. Then $\{a\}\in\mathcal{P}(X)$. Since $\mathcal{P}(X)$ is transitive, $\{a\}\subseteq\mathcal{P}(X)$. Thus, $a\in\mathcal{P}(X)$. Hence, $a\subseteq X$. It follows that $X$ is transitive. ∎

Related:

CumulativeHierarchy

Major Section:

Reference

Type of Math Object:

Theorem

Parent:

## Mathematics Subject Classification

03E20*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Comments

## gap in my education

I know no transitive sets. Please give a concrete example!

## Re: gap in my education

pahio writes:

> I know no transitive sets.

> Please give a concrete example!

The empty set!

More generally, any ordinal is a transitive set, if we use von Neumann's definition of ordinals (as we usually do): http://planetmath.org/encyclopedia/VonNeumannOrdinal.html

## Re: gap in my education

Some concrete examples are given in the entry with canonical name NaturalNumber. It gives the definition of 0, 1, 2, and 3 as ordinals, and indicates a pattern of how to form the rest.

You may also want to see the entry with canonical name VonNeumannOrdinal.

## Re: gap in my education

Tahanks Warren! The natural numbers are sufficiently simple for me =o)

Jussi

## Re: gap in my education

On the same topic, I have filed a request for an entry that gives an example of a transitive set that is not an ordinal. I do not know of any such sets.

## Re: gap in my education

I've fulfilled this request with an entry about the cumulative hierarchy: http://planetmath.org/encyclopedia/CumulativeHierarchy.html

However, your own entry (to which this thread is attached) also gives examples of such sets, since the power set of an ordinal > 1 is not an ordinal.