The Empty Set

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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.

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