Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

View the Project on GitHub neugierde/cantors-attic

**Quick navigation**

The upper attic

The middle attic

The lower attic

The parlour

The playroom

The library

The cellar

**Sources**

Cantor's Attic (original site)

Joel David Hamkins blog post about the Attic

Latest working snapshot at the wayback machine

A set is transitive if and only if all of its elements are subsets.

Equivalently, a set $A$ is transitive if and only if:

- it contains its union
- the powerset of $A$ contains $A$
- all members of the members of $A$ are members of $A$

If $A$ is transitive, then if $x$ and $A$ are connected somehow by membership (that is, $x \in y \in z \ldots \in A$), then $x \in A$.

The intersection of two transitive sets is transitive.

In set theory, transitive sets play an important role in models of ZFC. See transitive ZFC model.