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\begin{document}

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\section*{numeral}

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

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{arithmetic}{}\paragraph*{{Arithmetic}}\label{arithmetic}

[[!include arithmetic geometry - contents]]

\hypertarget{numerals}{}\section*{{Numerals}}\label{numerals}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{canonicity}{Canonicity}\dotfill \pageref*{canonicity} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In informal mathematical speech and writing, a \textbf{numeral} refers to any notation or terms which denotes a [[natural number]] directly. Usually, in mathematics, this refers to base ten place-value notation, so that for instance $0$, $2$, $13$, and $890$ are numerals.

In the more formal world of [[logic]] and [[type theory]], a \textbf{numeral} generally means (e.g. \hyperlink{MartinLoef73}{Martin-L\"o{}f 73}) a [[term]] whose type is a [[natural numbers type]] and which is of [[canonical form]]. The meaning of ``canonical form'' may vary with the formal theory, but with the usual presentation of the [[natural numbers]] as an [[inductive type]] generated by [[zero]] $0$ and the [[successor]] operation $s$, the numerals are the terms of the form

\begin{displaymath}
s(s(\cdots s(s(0))\cdots )).
\end{displaymath}
Often, the numeral representing the natural number $n$ --- which is to say, the term with $s$ applied $n$ times to $0$ --- is denoted by $\underline{n}$. Thus, for instance, $\underline{2}$ means $s(s(0))$. It is important to note that $\underline{2}$ is not (usually) a term \emph{inside} the formal system being considered; it is a ``meta-notation'' which stands for the term $s(s(0))$. (One might say that the underline converts \emph{informal} numerals to \emph{formal} ones.) In particular, any statement which quantifies over a natural number $n$ that occurs in a term $\underline{n}$ can only be expressed in the [[metatheory]].

\hypertarget{canonicity}{}\subsection*{{Canonicity}}\label{canonicity}

Not every term of natural number type $\mathbb{N}$ is a numeral; consider for instance $\underline{2}+\underline{2}$. However, good formal systems have the property of [[canonicity]], which in this context means that every term of type $\mathbb{N}$ \emph{computes to}, or is \emph{provably [[equal]] to}, a numeral. In our example, if $+$ is defined by [[recursion]], there is a sequence of [[beta-reduction]] steps leading from $\underline{2}+\underline{2}$ to $\underline{4}$. (Canonicity is about terms in the [[empty context]]; in a [[context]] with [[free variables]] of type $\mathbb{N}$, then of course there will be more terms of type $\mathbb{N}$, built out of these variables.)

If we add to such a formal system an [[axiom]] using an [[existential quantification|existential statement]], then this is equivalent to adding to the language an additional term for a natural number that is not (and may not provably be equal to) any canonical numeral. For example, in $PA + \neg{Con(PA)}$ ([[Peano arithmetic]] plus the axiom that Peano arithmetic is [[inconsistent]]), we have the axiom

\begin{displaymath}
\exists n, (n \vdash_{PA} \bot) ,
\end{displaymath}
stating the existence of a number $n$ that is the [[Goedel numbering|Gödel number]] of a [[proof]] in $PA$ of a [[falsehood]]. If we instead extend the language of $PA$ with a new symbol $&#8251;$ and add the axiom

\begin{displaymath}
&#8251; \vdash_{PA} \bot ,
\end{displaymath}
then (assuming that $PA$ and so $PA + \neg{Con(PA)}$ is in fact consistent) one can prove

\begin{displaymath}
\underline{n} \neq &#8251;
\end{displaymath}
in this system for every natural number $n$, but one cannot prove

\begin{displaymath}
\forall n,\, n \neq &#8251;
\end{displaymath}
(which is actually trivially refutable).

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

\begin{itemize}%
\item [[number]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

In

\begin{itemize}%
\item [[Per Martin-Löf]], \emph{An intuitionistic theory of types: predicative part}, (1973)

\end{itemize}
it says

\begin{quote}%
A closed normal term with type symbol $N$, which obviously must have the form $s(s(...s(0)...))$, is called a \emph{numeral}.


\end{quote}
[[!redirects numeral]] [[!redirects numerals]]



\end{document}