Definition:Transitive Class

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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

  • Results about transitive classes can be found here.


Sources

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