Definition:Empty Set

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The empty set is a set which has no elements.

It is usually denoted by some variant of a zero with a line through it, for example $\varnothing$ or $\empty$, and can always be represented as $\left\{{}\right\}$.

Axiomatic Set Theory

The concept of the empty set is axiomatised in the Axiom of the Empty Set in Zermelo-Fraenkel set theory:

There exists a set that has no elements:

$\exists x: \forall y: \left({\neg \left({y \in x}\right)}\right)$

Also known as

The empty set is sometimes called the null set, but this name is discouraged because there is another concept for null set which ought not to be confused with this.

Some sources call the empty set the vacuous set.

Others call it the void set.

Notes on Symbology

The symbols $\varnothing$ and $\empty$ are properly considered as stylings of $0$ (zero), and not variants of the Greek "Phi": $\Phi, \phi, \varphi$.

Some sources maintain that it is a variant on the Norwegian / Danish / Faeroese letter Ø.

Some sources use $\Box$ as the symbol for the empty set, but this is rare.

Other sources use $0$, but this is not recommended for readily apparent reasons.

The preferred symbol on $\mathsf{Pr} \infty \mathsf{fWiki}$ is $\varnothing$ for its completely unambiguous interpretation and aesthetically pleasing, clean presentation.

Existence of Empty Set

Some authors have problems with the existence (or not) of the empty set:

One may regard [the vacuous set] as a zero element that is adjoined to the collection of "real" subsets.
If $A, B$ are disjoint, then $A \cap B$ is not really defined, because it has no elements. For this reason we introduce a conventional empty set, denoted $\varnothing$, to be thought of as a 'set with no elements'. Of course this is a set only by courtesy, but it is convenient to allow $\varnothing$ the status of a set.
The best attitude towards the empty set $\varnothing$ is, perhaps, to regard it as an interesting curiosity, a convenient fiction. To say that $x \in \varnothing$ simply means that $x$ does not exist. Note that it is conveniently agreed that $\varnothing$ is a subset of every set, for elements of $\varnothing$ are supposed to possess every property.
Now some students are bothered with the notion of an "empty set". "How", they say, "can you have a set with nothing in it?" ... The empty set is only a convention, and mathematics could very well get along without it. But it is a very convenient convention, for it saves us a good deal of awkwardness in stating theorems and proving them.

Such a philosophical position is considered by many mathematicians to be a timid attitude harking back to the mediaeval distrust of zero.

Other sources allow the definition of the empty set, but because of the way natural numbers are defined, determine that it is neither finite nor infinite.

Historical Note

The concept of the empty set was stated by Leibniz in his initial conception of symbolic logic.

The use of $\varnothing$ has relevance to the early days of the computing industry, when $\empty$ was frequently used to mean "zero", in order to distinguish it from $\mathrm O$ (the letter O).

In the same context, the letter O was sometimes seen rendered as $\odot$, so as to ensure its being differentiated from "zero".

The latter has fallen out of use, but it is still common for mathematicians, when writing their mathematics by hand, to strike through their zeroes out of habit.

Also see

  • Empty Set is Unique for a proof that it is justifiable to refer to $\varnothing$ as the empty set.
  • Results about the empty set can be found here.