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
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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:
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.