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\begin{document}

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\section*{fan theorem}

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

\hypertarget{analysis}{}\paragraph*{{Analysis}}\label{analysis}

[[!include analysis - contents]]

\hypertarget{the_fan_theorem}{}\section*{{The fan theorem}}\label{the_fan_theorem}

\noindent\hyperlink{introduction}{Introduction}\dotfill \pageref*{introduction} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{obfuscation}{Obfuscation}\dotfill \pageref*{obfuscation} \linebreak
\noindent\hyperlink{variations}{Variations}\dotfill \pageref*{variations} \linebreak
\noindent\hyperlink{use_in_analysis}{Use in analysis}\dotfill \pageref*{use_in_analysis} \linebreak
\noindent\hyperlink{analysis_without_the_fan_theorem}{Analysis without the fan theorem}\dotfill \pageref*{analysis_without_the_fan_theorem} \linebreak
\noindent\hyperlink{sheaf_models}{Sheaf models}\dotfill \pageref*{sheaf_models} \linebreak
\noindent\hyperlink{proofs}{Proofs}\dotfill \pageref*{proofs} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{introduction}{}\subsection*{{Introduction}}\label{introduction}

The fan theorem is one of the basic principles of [[intuitionism]] that make it more specific (even in [[mathematics|mathematical]] practice, independent of any [[philosophy|philosophical]] issues) than garden-variety [[constructive mathematics]]. Its main use is to justify pointwise [[analysis]]; without it, one really needs [[locale theory]] for [[point-free topology]] instead. In [[classical mathematics]], the fan theorem is [[true]].

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

Consider the [[list|finite]] and [[infinite sequence|infinite]] sequences of [[binary digits]]. Given an infinite sequence $\alpha$ and a [[natural number]] $n$, let $\bar \alpha n$ be the finite sequence consisting of the first $n$ [[elements]] of $\alpha$.

Let $B$ be a collection of [[finite set|finite]] sequences of bits (or \emph{bitlists}), that is a [[subset]] of the [[free monoid]] on the [[boolean domain]]. Given an infinite sequence $\alpha$ and a natural number $n$, we say that $\alpha$ \emph{$n$-bars} $B$ if $\bar \alpha n \in B$; given only $\alpha$, we say that $\alpha$ \emph{bars} $B$ if $\alpha$ $n$-bars $B$ for some $n$.

We are interested in these three properties of $B$:

\begin{itemize}%
\item $B$ is \emph{[[decidable subset|decidable]]}: For every finite sequence $u$, either $u \in B$ or $u \notin B$. (This is trivial in [[classical logic]] but will hold constructively only for some subsets $B$.)
\item $B$ is \emph{barred}: For every infinite sequence $\alpha$, $\alpha$ bars $B$.
\item $B$ is \emph{uniform}: For some natural number $M$, for every infinite sequence $\alpha$, if $\alpha$ bars $B$ at all, then $\alpha$ $n$-bars $B$ for some $n \leq M$.

\end{itemize}
A \textbf{bar} is a barred subset $B$.

\begin{utheorem}
Every decidable bar is uniform. (In other words, if a collection of bitlists is decidable and barred, then it is also uniform.)

\end{utheorem}
Although the fan theorem is about bars, it is different from the [[bar theorem]], which is related but stronger.

\hypertarget{obfuscation}{}\subsubsection*{{Obfuscation}}\label{obfuscation}

Let $\mathbb{B}$ be the set $\{0,1\}$ of binary digits (bits) and $\mathbb{N}$ the set $\{0,1,2,\ldots\}$ of natural numbers (numbers). Given a [[set]] $A$, let $A^*$ be the set of finite sequences of elements of $A$, let $A^{\mathbb{N}}$ be the set of infinite sequences of elements of $A$, and let $\mathcal{P}_{\Delta}A$ be the set of decidable subsets of $A$. Then the fan theorem is about (elements of) $\mathbb{B}^*$, $\mathbb{B}^{\mathbb{N}}$, and $\mathcal{P}_{\Delta}\mathbb{B}^*$.

However, the sets $\mathbb{N}$, $\mathbb{B}^*$, and $\mathbb{N}^*$ are all isomorphic. Similarly, the sets $\mathbb{B}^{\mathbb{N}}$, $\mathcal{P}_{\Delta}\mathbb{N}$, $\mathcal{P}_{\Delta}\mathbb{B}^*$, and $\mathcal{P}_{\Delta}\mathbb{N}^*$ are all isomorphic. In much of the literature on bars, one tacitly uses all of these isomorphisms, taking $\mathbb{N}$ and $\mathbb{B}^{\mathbb{N}}$ as chosen representatives of their isomorphism classes. Thus, everything in sight is either a natural number or an infinite sequence of bits.

The fan theorem is hard enough to understand when $\alpha$ is an infinite sequence of bits and $\bar \alpha n$ is a finite sequence of bits; it is even harder to understand when $\bar \alpha n$ is a natural number that bears no immediate relationship to the digits in the sequence $\alpha$.

\hypertarget{variations}{}\subsubsection*{{Variations}}\label{variations}

The fan theorem may be stated about \emph{all} bars, not just the decidable ones: all bars are uniform (which is true in classical mathematics). Brouwer himself at one point claimed this, but later Kleene showed that this contradicted [[L.E.J. Brouwer|Brouwer]]'s [[continuity theorem]].

Since decidability is classically trivial, we may call this the \textbf{classical fan theorem}.

\hypertarget{use_in_analysis}{}\subsection*{{Use in analysis}}\label{use_in_analysis}

In [[classical mathematics]], the fan theorem is simply true.

In [[constructive mathematics]], the fan theorem is equivalent to any and all of the following statements:

\begin{itemize}%
\item As a [[locale]], [[Cantor space]] has enough points (is [[topological locale|topological]]).
\item As a [[topological space]], Cantor space is [[compact space|compact]].
\item As a topological space, the (located Dedekind) [[unit interval]] is [[compact space|compact]] (the [[Heine-Borel theorem|Heine–Borel theorem]]).
\item As a topological space, the (located Dedekind) [[real line]] is [[locally compact space|locally compact]].
\item Every [[uniformly continuous map|uniformly continuous]] function from Cantor space to the [[metric space]] $\dot{\mathbb{R}}^+$ of positive real numbers has a positive lower bound.
\item Every [[uniformly continuous map|uniformly continuous]] function from the unit interval to $\dot{\mathbb{R}}^+$ has a positive lower bound.
\item (This item might need dependent choice in order to be equivalent to the others.) There exists a class of ``kontinuous'' [[partial functions]] from the set $\mathbb{R}$ of (located Dedekind) [[real numbers]] to itself (see Waaldijk) such that\begin{itemize}%
\item the [[restriction]] of a kontinuous function to any smaller domain is kontinuous;
\item the [[identity function]] on $\mathbb{R}$ is kontinuous;
\item the [[composite]] of two kontinuous functions is kontinuous;
\item a function whose domain is the unit interval is kontinuous \emph{if and only if} it is uniformly continuous (in the usual metric-space sense); and
\item the function $(x \mapsto 1/x)$ defined on $\dot{\mathbb{R}}^+$ is kontinuous.

\end{itemize}


\end{itemize}
It follows from any of these statements:

\begin{itemize}%
\item The [[bar theorem]] holds.
\item As a locale, [[Baire space of irrational numbers|Baire space]] has enough points.
\item Every pointwise-continuous function on Cantor space is uniformly continuous.
\item Every pointwise-continuous function on the unit interval is uniformly continuous.
\item Every [[uniformity]] (or indeed any [[metric]]) compatible with the usual topology of Cantor space is [[totally bounded space|totally bounded]].
\item The property of being ``complete and totally bounded'', i.e. [[Bishop-compact]], is invariant under topological homemorphism between uniform (or metric) spaces.

\end{itemize}
I need to figure out how it relates to the various versions of [[Konig's lemma|König's Lemma]], as well as these statements (which are mutually equivalent):

\begin{itemize}%
\item As a locale, the unit interval has enough points.
\item As a locale, the [[real line]] (the [[locale of real numbers]]) has enough points.

\end{itemize}
Some of the results above may use [[countable choice]], but probably no more than $AC_{0,0}$ (which is choice for relations between $\mathbb{N}$ and itself).

\hypertarget{analysis_without_the_fan_theorem}{}\subsubsection*{{Analysis without the fan theorem}}\label{analysis_without_the_fan_theorem}

Point-wise real analysis without the fan theorem is very difficult, as shown by the example above from Waaldijk regarding ``kontinuous'' functions: without the fan theorem there isn't really even a good notion of continuity! This was [[L.E.J. Brouwer|Brouwer]]'s motivation for introducing the fan theorem.

However, the fan theorem (and bar theorem) can be avoided by instead using [[locales]] or another [[point-free topology|point-free]] approach to analysis.

\hypertarget{sheaf_models}{}\subsubsection*{{Sheaf models}}\label{sheaf_models}

\hyperlink{FourmanHyland}{Fourman and Hyland} provide a sheaf model not satisfying the fan theorem.

\hypertarget{proofs}{}\subsection*{{Proofs}}\label{proofs}

I should write down the classical proof (which uses [[excluded middle]] and some form of [[dependent choice]]), as well as Brouwer's argument.

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

\begin{itemize}%
\item Thanks to Giovanni Curi on \href{http://groups.google.com/group/constructivenews/browse_thread/thread/9d57fa99e67e8782}{constructive news}.


\item \href{http://www.fwaaldijk.nl/mathematics.html}{Frank Waaldijk} pointed out exactly why point-wise analysis needs the fan theorem.



\end{itemize}
I need to read the relevant parts here:

\begin{itemize}%
\item [[Mike Fourman]], R. Grayson, \emph{Formal Spaces}. In: The L.E.J. Brouwer Centenary Symposium, A.S. Troelstra and D. van Dalen, eds. North Holland (1982), pp. 107--122.

\end{itemize}
More links that I need to keep in mind:

\begin{itemize}%
\item \href{http://golem.ph.utexas.edu/category/2008/12/the_status_of_coalgebra.html}{http\char58\char47\char47golem\char46ph\char46utexas\char46edu\char47category\char47\char50\char48\char48\char56\char47\char49\char50\char47the\char95status\char95of\char95coalgebra\char46html}
\item \href{http://www.jaist.ac.jp/is/labs/ishihara-lab/wcalm2010/berger.pdf}{http\char58\char47\char47www\char46jaist\char46ac\char46jp\char47is\char47labs\char47ishihara\char45lab\char47wcalm\char50\char48\char49\char48\char47berger\char46pdf}
\item \href{http://www.cairn.info/revue-internationale-de-philosophie-2004-4-page-483.htm}{http\char58\char47\char47www\char46cairn\char46info\char47revue\char45internationale\char45de\char45philosophie\char45\char50\char48\char48\char52\char45\char52\char45page\char45\char52\char56\char51\char46htm}

\end{itemize}
Also:

\begin{itemize}%
\item [[Mike Fourman]], [[Martin Hyland]], \emph{Sheaf models for analysis}. \href{https://www.dpmms.cam.ac.uk/~martin/Research/Oldpapers/analysis79.pdf}{PDF}

\end{itemize}
[[!redirects fan theorem]] [[!redirects Fan theorem]]

[[!redirects classical fan theorem]]



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