# Definition:Transitive Class

## Definition

Let $A$ denote a class, which can be either a set or a proper class.

Then $A$ is **transitive** if and only if every element of $A$ is also a subclass of $A$.

That is, $A$ is **transitive** if and only if:

- $x \in A \implies x \subseteq A$

or:

- $\forall x: \forall y: \paren {x \in y \land y \in A \implies x \in A}$

## Notation

In order to indicate that a class $A$ is **transitive**, this notation is often seen:

- $\operatorname {Tr} A$

whose meaning is:

**$A$ is (a) transitive (class or set).**

Thus $\operatorname {Tr}$ can be used as a propositional function whose domain is the class of all classes.

## Also known as

A **transitive class** is also known as a **complete class**.

Thus a class which is not **transitive** can be considered to be a class with "holes" in it.

## Examples

### Empty Class

The empty class is transitive.

### Singleton of Empty Class

Let $\O$ denote the empty class.

Then the singleton $\set \O$ is transitive.

### Class $\set {\O, \set \O}$

Let $\O$ denote the empty class.

Then the class:

- $\set {\O, \set \O}$

is transitive.

### Class $\set {\O, \set \O, \set {\O, \set \O} }$

Let $\O$ denote the empty class.

Consider the ordinal $3$, defined as:

- $\mathcal 3 := \set {\O, \set \O, \set {\O, \set \O} }$

$\mathcal 3$ is transitive.

### Class $\set {\O, \set {\O, \set \O} }$

Let $\O$ denote the empty class.

Consider the class $S$, defined as:

- $S := \set {\O, \set {\O, \set \O} }$

$S$ is not transitive.

### Singleton of Singleton of Empty Class is not Transitive

Let $\O$ denote the empty set.

Consider the class $S$, defined as:

- $S := \set {\set \O}$

$S$ is not transitive.

## Also see

- Class has Subclass which is not Element: while there is nothing stopping a class to be such that all its elements are subclasses, it is not possible for
*all*its subclasses to be among its elements.

- Results about
**transitive classes**can be found here.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 12$: The Peano Axioms - 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 7.1$ - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 2$ Transitivity and supercompleteness: Definition $2.1$