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 $2.1$