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.


Also see

  • Results about transitive classes can be found here.


Also see


Sources