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\begin{document}

%-------------------------------------------------------------------


\section*{empty set}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{foundational_status}{Foundational status}\dotfill \pageref*{foundational_status} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The \textbf{empty set} $\emptyset$ is the [[set]] with no [[element]]s.

The empty set is unique in most membership-based [[foundations]], but in any case we can say `the' empty set, since any two empty sets are isomorphic by a unique isomorphism (the [[generalized the]]).

Indeed, in any foundations (whether membership-based or structural), the empty set is characterized by a [[universal property]]: it is the [[initial object]] in the [[category]] [[Set]] of all sets: for every set $S$, there is a unique [[function]]

\begin{displaymath}
\emptyset \stackrel{\exists !}{\to} S
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

The empty set can be confusing, because it is a degenerate case. Indeed, it is defined as an exception: every set is [[inhabited set|inhabited]], except the empty set. Nothing belongs to the empty set, but the empty set itself is something. Mathematics students are confused that there is a function from the empty set to itself, and even grown mathematicians will misstate definitions because they forget about the empty set.

In terms of the empty set one can give sense to the [[natural number object|natural number]] expression $0^0$, the [[exponentiation]] of [[zero]] with itself, by defining the [[cartesian closed category|exponential]] $m^n$ to be the [[cardinality]] of the [[hom-set]] $\hom_{Set}([n], [m])$ where $[m]$ is an $m$-element set. Then $0^0 = 1$ because $\emptyset$ is an [[initial object]].

\begin{remark}
\label{}\hypertarget{}{}
Endless debates (as seen for example at sci.math) about the meaning of $0^0$ are most easily resolved by paying attention to how exponentiation (in a [[ring]] or [[rig]] that admits [[exponentiation]]) is defined. We have just observed how $0^0 = 1$ makes good sense in the rig of natural numbers. But exponentiation for the [[real number]] field ($x^y$ for $x \gt 0$) is standardly defined by continuity considerations, and cannot be continuously extended to cover $0^0$. Of course, that has nothing to do with the empty set!

\end{remark}
\hypertarget{foundational_status}{}\subsection*{{Foundational status}}\label{foundational_status}

In [[set theory]] as a [[foundation of mathematics]], the existence of the empty set is often taken as an axiom, the \textbf{axiom of the null set}. Actually, this axiom can be written in such a simple way that the name seems unmotivated:

\begin{udefn}
There exists a set.

\end{udefn}
However, one can then use the axiom of separation ([[bounded separation]] is enough) to prove that there exists an empty set, and then use the [[axiom of extensionality]] to prove that this empty set is unique.

In some axiomatic frameworks, even this is unnecessary; while it makes little sense from the standpoint of [[categorial logic]], it was once traditional to take $\forall x,\; P[x] \;\Rightarrow\; \exists x,\; P[x]$ as an axiom (or theorem) of [[first-order logic]]. It is then automatic that something exists, and if one uses a [[material set theory]] of [[pure sets]], then this something must be a set.

Alternatively, since the [[axiom of infinity]] states the existence of a set (and is often phrased to state explicitly the existence of an empty set), then the axiom of the null set becomes unnecessary for another reason.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[!include empty objects -- contents]]

\begin{itemize}%
\item [[nothing]]

\end{itemize}
category: foundational axiom

[[!redirects empty set]] [[!redirects empty sets]]

[[!redirects axiom of empty set]] [[!redirects axiom of the empty set]] [[!redirects axiom of null set]] [[!redirects axiom of the null set]]



\end{document}