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

The empty set (denoted $\emptyset$ or $\varnothing$) is the only set
$S$ such that $\neg\exists a(a\in S)$. It contains absolutely no
elements, and has cardinality 0. It is often thought of to be the only
urelement
(this holds up in $V$), and is *increadibly* important as a result. It
is also one of the only ranks to also be an ordinal, and contains many
properties when put in a poset.

The empty set is ordered by every relation, not concerning any urelements. When ordered by any relation at all, the empty poset is every one of the following:

- Transitive
- Reflexive
- A total order

The Von Neumann ordinal $\varnothing$ is 0, and is the only ordinal equivalent to it’s own rank, other than $V_1=1=\{\varnothing\}$ and $V_2=2=\{\varnothing,\{\varnothing\}\}$. There is some debate whether or not it is a limit ordinal.

The empty function is relatively uninteresting. It’s domain and range are both $\varnothing$ (and thus it is a bijective function). For this reason it is often considered trivial. However, the empty relation has many, many properties that could only be attributed to itself. It is the only relation that is all of the following:

- Transitive
- Reflexive, Irreflexive, and Coreflexive (The only relation where this is true)
- A total order
- Symmetric, Assymetric, and Antisymmetric (The only relation where this is true)
- An equivalence relation
- Trichotomous
- Euclidean
- Serial
- Set-Like