You signed in with another tab or window. Reload to refresh your session.You signed out in another tab or window. Reload to refresh your session.You switched accounts on another tab or window. Reload to refresh your session.Dismiss alert
Copy file name to clipboardExpand all lines: doc/ordinal-chapter.tex
+23-16Lines changed: 23 additions & 16 deletions
Original file line number
Diff line number
Diff line change
@@ -46,8 +46,7 @@ \chapter{Introduction to ordinal numbers and ordinal notations}
46
46
\item Dots : ``$\ldots$'' stand for infinite sequences of ordinals, not shown for lack of space. For instance, the ordinal $42$ is not shown in the first line, but it exists, somewhere between $17$ and $\omega$.
47
47
\item Each ordinal printed in black is the immediate successor of another ordinal. We call it a \emph{successor} ordinal. For instance, $12$ is the successor of $11$, and $\omega^4+1$ the successor of $\omega^4$.
48
48
\item Ordinals (displayed in red) that follow immediately dots are called \emph{limit ordinals}. With respect to the order induced by this sequence, any limit ordinal $\alpha$ is the least upper bound of the set $\mathbb{O}_\alpha$ of all ordinals strictly less than $\alpha$.
49
-
\item
50
-
For instance $\omega$ is the least upper bound of the set of all finite ordinals (in the first line). It is also the first limit ordinal, and the first infinite ordinal, in the sense that
49
+
For instance,$\omega$ is the least upper bound of the set of all finite ordinals (in the first line). It is also the first limit ordinal, and the first \emph{infinite ordinal number}, in the sense that
51
50
the set $\mathbb{O}_\omega$ is infinite.
52
51
\item The ordinal $\epsilon_0$ is the first number which is equal to its own exponential of base $\omega$. It plays an important role in proof theory, and is particularly studied in chapters~\ref{chap:T1} to \ref{chap:alpha-large}.
53
52
\item Any ordinal is either the ordinal \textcolor{blue}{$0$},
@@ -59,6 +58,15 @@ \chapter{Introduction to ordinal numbers and ordinal notations}
59
58
60
59
\section{The mathematical point of view}
61
60
61
+
62
+
63
+
We cannot cite all the literature published on ordinals since Cantor's book
64
+
\cite{cantorbook}, and
65
+
leave it to the reader to explore the bibliography.
66
+
The introduction of Jos\'e Grimm's report~\cite{grimm:hal-00911710} contains also a nice presentation of the main properties of ordinals.
67
+
68
+
For simplicity's sake, we will only give the definitions which are useful for understanding our \coq development.
69
+
62
70
\subsection{Well-ordered sets}
63
71
Let us start with some definitions.
64
72
A \emph{well-ordered set} is a set provided with a binary relation $<$ which has the following properties.
@@ -88,21 +96,15 @@ \subsection{Ordinal numbers}
88
96
there exists a strictly monotonous bijection $b$ from $A$ to $B$, \emph{i.e.} which verifies the proposition
89
97
$\forall x\,y\in A,\, x <_A y \Rightarrow b(x) <_B b(y)$.
90
98
91
-
Having the same order type is an equivalence relation between well-ordered sets. Ordinal numbers (in short: \emph{ordinals}) are descriptions (\emph{names}) of the equivalence classes.
99
+
Having the same order type is an equivalence relation between well-ordered sets. Ordinal numbers (in short: \emph{ordinals}) are descriptions (\emph{names}) of the associated equivalence classes.
92
100
For instance, the order type of $(\mathbb{N},<)$ is associated with the ordinal called $\omega$, and the order we considered on
93
-
the disjoint union of $\mathbb{N}$and itself is named$\omega+\omega$.
101
+
the disjoint union of two copies of $\mathbb{N}$is associated with $\omega\times2$ (a.k.a. $\omega+\omega$).
94
102
95
103
In a set-theoretic framework, one can consider any ordinal $\alpha$ as a well-ordered set, whose elements are just the ordinals strictly less than $\alpha$, \emph{i.e.} the \emph{segment} $\mathbb{O}_\alpha=[0, \alpha)$. So, one can speak about \emph{finite}, \emph{infinite}, \emph{countable}, etc., ordinals. Nevertheless, since we work within type theory,
96
104
we do not identify ordinals as sets of ordinals, but consider the correspondence between ordinals and sets of ordinals as the function that maps $\alpha$ to $\mathbb{O}_\alpha$.
97
105
For instance $\mathbb{O}_\omega=\mathbb{N}$, and $\mathbb{O}_7=\{0,1,2,3,4,5,6\}$.
98
106
99
107
100
-
We cannot cite all the literature published on ordinals since Cantor's book
101
-
\cite{cantorbook}, and
102
-
leave it to the reader to explore the bibliography.
103
-
104
-
The introduction of Jos\'e Grimm's report~\cite{grimm:hal-00911710} contains a nice presentation of the main properties of ordinals.
105
-
106
108
107
109
\section{Ordinal numbers in Coq}
108
110
@@ -139,7 +141,7 @@ \section{Ordinal Notations}
139
141
Fortunately, the ordinals we need for studying hydra battles are much simpler than Schütte's, and can be represented as quite simple data types in \gallina.
140
142
141
143
Let $\alpha$ be some (countable) ordinal;
142
-
in \coq{} terms, we call \emph{ordinal notation for $\alpha$} a structure composed
144
+
we call \emph{ordinal notation for $\alpha$} any structure composed
143
145
of:
144
146
\begin{itemize}
145
147
\item A data type $A$ for representing all ordinals strictly below $\alpha$,
The \texttt{Comparable} class, contributed by Jérémy Damour and Théo Zimmermann, allows us to apply generic lemmas and tactics about decidable strict orders.
165
167
The correctness of the comparison function is expressed through Stdlib's type
166
-
\texttt{Datatypes.CompareSpec} as specialized by\texttt{Datatypes.CompSpec}.
168
+
\texttt{Datatypes.CompareSpec} and predicate\texttt{Datatypes.CompSpec}.
167
169
168
170
\begin{Coqsrc}
169
171
Inductive CompareSpec (Peq Plt Pgt : Prop) :
@@ -291,13 +293,16 @@ \section{Sum of two ordinal notations}
291
293
Standard library's lemma \texttt{Wellfounded.Disjoint\_Union.wf\_disjoint\_sum}
292
294
is applied to prove that our order \texttt{lt} is well-founded, allowing us to build an instance of \texttt{ON}:
0 commit comments