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\begin{document}

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\section*{natural number}

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

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

[[!include arithmetic geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{natural_numbers_objects}{Natural numbers objects}\dotfill \pageref*{natural_numbers_objects} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{minima_of_subsets_of_natural_numbers}{Minima of subsets of natural numbers}\dotfill \pageref*{minima_of_subsets_of_natural_numbers} \linebreak
\noindent\hyperlink{decreasing_sequences_of_natural_numbers}{Decreasing sequences of natural numbers}\dotfill \pageref*{decreasing_sequences_of_natural_numbers} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

A \textbf{natural number} is traditionally one of the [[numbers]] $1$, $2$, $3$, and so on. It is now common in many fields of mathematics to include $0$ as a natural number as well. One advantage of doing so is that a natural number can then be identified with the [[cardinal number|cardinality]] of a [[finite set]], as well as a finite [[ordinal number]]. One can distinguish these as the \textbf{nonnegative integers} (with $0$) and the \textbf{positive integers} (without $0$), at least until somebody uses `positive' in the semidefinite sense. To a [[set theory|set theorist]], a natural number is essentially the same as an \textbf{[[integer]]}, so they often use the shorter word; one can also clarify with \textbf{unsigned integer} (but this doesn't help with $0$).

The set of natural numbers is often written $N$, $\mathbf{N}$, $\mathbb{N}$, $\omega$, or $\aleph_0$. The last two notations refer to this set's structure as an [[ordinal number]] or [[cardinal number]] respectively, and they often (usually for $\aleph$) have a subscript $0$ allowing them to be generalised. In the [[foundations]] of mathematics, the [[axiom of infinity]] states that this actually forms a set (rather than a proper class). At a foundational level, it's completely irrelevant whether $0$ counts as a natural number or not; as [[sets]] (and even as [[natural numbers objects]]), the two options are equivalent, so we are really talking about the choice of [[rig]] structure (or [[inclusion map]] into the set of [[integers]], etc).

By default, our natural numbers always include $0$.

\hypertarget{natural_numbers_objects}{}\subsection*{{Natural numbers objects}}\label{natural_numbers_objects}

$\mathbf{N}$ is a [[natural numbers object]] in [[Set]]; indeed, it is the original example. This consists of an initial element $0$ (or $1$ if $0$ is not used) and a successor operation $n \mapsto n + 1$ (or simply $n \mapsto n^+$; in [[computer science]], one often writes $n+$) such that, for a set $X$, an element $a: X$, and a [[function]] $s: X \to X$, there exists a unique function $f: \mathbf{N} \to X$ such that $f_0 = a$ and $f_{n+} = s(f_n)$. This function $f$ is said to be constructed by \textbf{primitive recursion}. (Fancier forms of [[recursion]] are also possible.)

The basic idea is that we define the values of $f$ one by one, starting with $f_0 = a$, then $f_1 = s(a)$, $f_2 = s(s(a))$, and so on. These are all both possible and necessary individually, but something must be put in the [[foundations]] to ensure that this can go on uniquely forever.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{minima_of_subsets_of_natural_numbers}{}\subsubsection*{{Minima of subsets of natural numbers}}\label{minima_of_subsets_of_natural_numbers}

In [[classical mathematics]], any [[inhabited set|inhabited]] subset of the natural numbers possesses a minimal element. In [[constructive mathematics]], one cannot show this:

\begin{prop}
\label{}\hypertarget{}{}
\textbf{(a Brouwerian counterexample)}

If any inhabited subset of the natural numbers possesses a minimal element, then the [[excluded middle|law of excluded middle]] holds.

\end{prop}
\begin{proof}
Let $\varphi$ be an arbitrary arithmetical formula. Then the subset

\begin{displaymath}
U := \{ n \in \mathbb{N} \,|\, n = 1 \vee \varphi \} \subseteq \mathbb{N}
\end{displaymath}
is inhabited. By assumption, it possesses a minimal element $n_0$. By discreteness of the natural numbers, either $n_0 = 0$ or $n_0 \gt 0$. In the first case, $\varphi$ holds. In the second case, $\neg\varphi$ holds.

\end{proof}
In this sense, the natural numbers are not complete, and it's fruitful to study their completion: For instance, the global sections of the completed natural numbers object in the [[category of sheaves|sheaf topos]] on a topological space $X$ are in one-to-one correspondence with upper semicontinuous functions $X \to \mathbb{N}$ (details at \emph{[[one-sided real number]]}).

We can salvage the minimum principle in two ways:

\begin{prop}
\label{}\hypertarget{}{}
Any \textbf{[[decidable subset|detachable]]} inhabited subset of the natural numbers possesses a minimal element.

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
Any inhabited subset of the natural numbers does \textbf{not not} possess a minimal element.

\end{prop}
For instance, any finitely generated vector space over a [[field|residue field]] does \emph{not not} possess a finite basis (pick a minimal generating set, guaranteed to \emph{not not} exist). Interpreting this in the [[internal language]] of the sheaf topos of a [[reduced scheme]] $X$, one obtains the well-known fact that any $\mathcal{O}_X$-module locally of finite type over $X$ is locally free on a dense open subset.

\hypertarget{decreasing_sequences_of_natural_numbers}{}\subsubsection*{{Decreasing sequences of natural numbers}}\label{decreasing_sequences_of_natural_numbers}

Classically, any \emph{weakly} decreasing sequence of natural numbers $(a_n)_n$ is eventually constant, i.e. admits an index $N$ such that $a_N = a_{N+1} = a_{N+2} = \cdots$. Constructively, one can only prove for each $M$ that there exists an index $N$ such that $a_N = a_{N+1} = \cdots = a_{N+M}$. (One may prove this by induction on $a_0$; indeed, you can always find $N$ so that $N \leq a_0 M$.) The classical principle is equivalent to the [[limited principle of omniscience]] for $\mathbb{N}$ (which follows already when $a_0 = 1$).

On the other hand, there can be no \emph{strictly} decreasing sequence of natural numbers. This is constuctively valid (proved by contradiction and induction on $a_0$).

This is relevant to [[constructive algebra]], as this shows that formulating chain conditions needs some care. (It is easier to say `weakly' than `strictly' in the hypothesis, but then it's unclear how to state the conclusion.)

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

\begin{itemize}%
\item [[number]], [[natural number]], [[integer]], [[rational number]], [[algebraic number]], [[Gaussian number]], [[irrational number]], [[real number]], [[p-adic number]]


\item [[natural numbers type]], [[natural numbers object]]


\item [[carrying]]


\item [[numeral]]


\item [[countable ordinal]]



\end{itemize}
[[!redirects natural number]] [[!redirects natural numbers]]

[[!redirects nonnegative integer]] [[!redirects nonnegative integers]] [[!redirects positive integer]] [[!redirects positive integers]] [[!redirects unsigned integer]] [[!redirects unsigned integers]]



\end{document}