# Definition
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
<a href="index.php?title=Urelement&action=edit&redlink=1" class="new" title="Urelement (page does not exist)">urelement</a>
(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.
## As 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
## As An Ordinal
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.
## As a Function\Relation
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