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\section*{well-order}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{$(0,1)$-Category theory}}\label{category_theory}

[[!include (0,1)-category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{wellorders_are_linear}{Well-orders are linear}\dotfill \pageref*{wellorders_are_linear} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{interpretation_as_an_ordinal_number}{Interpretation as an ordinal number}\dotfill \pageref*{interpretation_as_an_ordinal_number} \linebreak
\noindent\hyperlink{simulations}{Simulations}\dotfill \pageref*{simulations} \linebreak
\noindent\hyperlink{successor}{Successor}\dotfill \pageref*{successor} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{well-order} on a set $S$ is a [[relation]] $\prec$ that allows one to interpret $S$ as an [[ordinal number]] $\alpha$ and $\prec$ as the relation $\lt$ on the ordinal numbers less than $\alpha$. In particular, one can do [[induction]] on $S$ over $\prec$ (although the more general [[well-founded relations]] also allow this).

The \emph{[[well-ordering theorem]]} states precisely that every set may be equipped with a well-order. This theorem follows from the [[axiom of choice]], and is equivalent to it in the presence of [[excluded middle]].

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

A [[binary relation]] $\prec$ on a [[set]] $S$ is a \textbf{well-order} if it is [[transitive relation|transitive]], [[extensional relation|extensional]], and [[well-founded relation|well-founded]]. A set equipped with a well-order is called a \textbf{well-ordered set}, or (following `[[partial order|poset]]') a \textbf{woset}. Actually, the term `well-ordered' came first; `well-order' is a back formation, which explains the strange grammar.

Other definitions of a well-order may be found in the literature; they are equivalent given [[excluded middle]], but the definition above seems to be the most powerful in [[constructive mathematics]]. Specifically:

\begin{itemize}%
\item a well-order is precisely a [[well-founded relation|well-founded]] [[linear order]];
\item a well-order is precisely a [[well-founded relation|well-founded]] [[total order]];
\item (assuming also [[dependent choice]]) a well-order is precisely a linear order $\prec$ with no infinite descending sequence $\cdots \prec x_2 \prec x_1 \prec x_0$;
\item (assuming also [[dependent choice]]) a well-order is precisely a total order $\preceq$ such that every infinite descending sequence $\cdots \preceq x_2 \preceq x_1 \preceq x_0$ has $x_i = x_{i^+}$ for some $i$ (and hence for infinitely many $i$);
\item a well-order on $S$ is precisely a [[linear order]] $\prec$ with the property that every [[inhabited subset]] $U$ of $S$ has a least element (an element $\bot_U$ such that no $x \in U$ satisfies $x \prec \bot_U$;
\item a well-order on $S$ is precisely a [[total order]] $\preceq$ with the property that every [[inhabited subset]] $U$ of $S$ has a least element (an element $\bot_U$ such that every $x \in U$ satisfies $\bot_U \preceq x$.

\end{itemize}
The really interesting thing here is that every well-order is linear; it is a constructive theorem that every linear order is weakly extensional (and so extensional if well-founded) and transitive. (For a [[weak counterexample]], take the set of [[truth values]] with $x \prec y$ iff $y$ is true and $x$ is false; this is a well-order that's linear iff [[excluded middle]] holds.) For the other equivalences, we're simply using well-known classical equivalents for well-foundedness and the classical correspondence between a linear relation $\prec$ and its [[reflexive closure]] $\preceq$.

For reference, a \textbf{classical well-order} is any order satisfying the last definition (a total order that is classically well-founded). A classically well-ordered set is a [[choice object|choice set]], and so if any set with at least $2$ elements has a classical well-order, [[excluded middle]] follows.

\hypertarget{wellorders_are_linear}{}\subsubsection*{{Well-orders are linear}}\label{wellorders_are_linear}

As stated above, well-founded extensional transitive relations $\prec$ on a set $X$ are linear, assuming [[classical logic]].

\begin{proof}
Order $X \times X$ [[lexicographic order|lexicographically]]: $(a, b) \prec (a', b')$ if either $a \prec a'$ in $X$ or $a = a'$ and $b \prec b$ in $X$. It is not hard to see that the lexicographic order is well-founded (and in fact a well-order, although we do not need this). Now let $A \subset X \times X$ be the set of pairs $(x, y)$ of non-equal elements $x$ and $y$ that are incomparable in $X$, and suppose $A$ is inhabited. Then $A$ has a minimal element $(a, b)$ (using excluded middle). Then, for every $x \prec b$, either $a \preceq x$ or $x \preceq a$. If the former holds for some $x$, then $a \prec b$ follows by transitivity, contradiction. Hence $x \prec a$ for every $x \prec b$. Now let $a'$ be minimal such that $x \prec a' \preceq a$ for every $x \prec b$.

Claim:

\begin{displaymath}
\{x: x \prec a'\} = \{x: x \prec b\}.
\end{displaymath}
We know already the right side is contained in the left. In the other direction, suppose $x \prec a'$. Since $x \prec a$ and $(a, b)$ was chosen minimal in the lexicographic order, $x$ and $b$ are comparable. If $b \preceq x \prec a'$, this contradicts minimality of $a'$. Thus $x \prec b$, i.e., the left side is contained in the right.

But now, by extensionality, $a' = b$, whence $b \preceq a$, contradiction. Therefore $A$ was empty, so that $X$ is [[connected relation|connected]] and therefore (being already transitive and irreflexive and using excluded middle again) a linear order.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item Any [[finite set|finite]] [[linearly ordered set]] $\{x_1 \lt \cdots \lt x_n\}$ is well-ordered.


\item The set of [[natural numbers]] is well-ordered under the usual order $\lt$.


\item More generally, any set of [[ordinal numbers]] (or even the [[proper class]] of all ordinal numbers) is well-ordered under the usual order $\lt$ (which, constructively, may not be linear).


\item The [[cardinal numbers]] of well-orderable sets (the well-orderable cardinals), forming a [[retract]] of the ordinals, are well-ordered. So by the well-ordering theorem, the class of \emph{all} cardinal numbers is well-ordered.


\item A special case of the well-ordering theorem is the existence of a well-order on the set of [[real numbers]]; this is enough for many applications of the [[axiom of choice]] to [[analysis]].



\end{itemize}
\hypertarget{interpretation_as_an_ordinal_number}{}\subsection*{{Interpretation as an ordinal number}}\label{interpretation_as_an_ordinal_number}

Any well-ordered set $S$ defines an [[ordinal number]] $\alpha$ and an order isomorphism $r$ between $S$ and the set of ordinal numbers less than $\alpha$; as such, $S$ may be identified (up to isomorphism of wosets) with the von Neumann ordinal $\alpha$. The idea is that the minimal element $\bot$ of $S$ itself (if any) is mapped to the ordinal number $0$, the minimal element of $S \setminus \{\bot\}$ (if any) is mapped to $1$, and so on; after which the next element of $S$ (if any) is mapped to $\omega$, and so on; and so on.

This may be defined immediately (and constructively) as a recursively defined function from $S$ to the class of all ordinal numbers:

\begin{displaymath}
r(x) = \sup_{t \prec x} r(t)^+ ;
\end{displaymath}
the validity of this sort of recursive definition is precisely what the well-foundedness of $\prec$ allows. Here, $\beta^+$ is the [[successor]] of the ordinal number $\beta$, and $\sup$ is the [[supremum]] operation on ordinal numbers (which is the [[union]] of von Neumann ordinals).

Since $S$ is a set, the image of $r$ in the class of all ordinals is also a set (by the [[axiom of replacement]]), and one can now prove that $r$ is an order isomorphism between $S$ and the set of ordinals less than the next ordinal, $\alpha \coloneqq (\im r)^+$.

\hypertarget{simulations}{}\subsection*{{Simulations}}\label{simulations}

Given two well-ordered sets $S$ and $T$, a [[function]] $f\colon S \to T$ is a \textbf{[[simulation]]} of $S$ in $T$ if

\begin{itemize}%
\item $f(x) \prec f(y)$ whenever $x \prec y$ and
\item given $t \prec f(x)$, there exists $y \prec x$ with $t = f(y)$.

\end{itemize}
Note that any simulation of $S$ in $T$ must be unique. Thus, well-ordered sets and simulations form a category that is in fact a (large) [[preorder]], whose reflection in the category of [[posets]] is in fact the poset of [[ordinal numbers]].

\hypertarget{successor}{}\subsection*{{Successor}}\label{successor}

A well-ordered set $S$ comes equipped with a \textbf{[[successor]]}, which is a [[partial function|partial map]] $succ\colon S \to S  
  \,$ that sends $a \in S$ to the lowest element of the subset $S_a := \{ s \in S, a \prec s\}$, whenever this set is inhabited.

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{limit} well-order is a well-order $S$ whose successor map is a [[total function]].

\end{defn}
Similarly, one may define a successor [[functor]] on the [[category]] of well-ordered sets, taking $S$ to the well-order obtained by freely adjoining a (new) top element to $S$. Since this category (which is [[thin category|thin]]) can be regarded as itself a well-ordered [[proper class]], this is a special case of the successor operation above. (Hence the ordinal of all ordinals is a limit ordinal.)

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[well-quasi-order]]

\end{itemize}
[[!redirects well order]] [[!redirects well orders]] [[!redirects well-order]] [[!redirects well-orders]] [[!redirects well ordering]] [[!redirects well orderings]] [[!redirects well-ordering]] [[!redirects well-orderings]]

[[!redirects well ordered]] [[!redirects well-ordered]] [[!redirects well ordered set]] [[!redirects well ordered sets]] [[!redirects well-ordered set]] [[!redirects well-ordered sets]] [[!redirects woset]] [[!redirects wosets]]



\end{document}