History

History and Mythology of Set Theory

content/history/set-theory/set-theory.tex

% Part: history% Chapter: set-theory\documentclass[../../../include/open-logic-chapter]{subfiles}\begin{document}	\olchapter{his}{set}{History and Mythology of Set Theory}\begin{editorial}This chapter includes the historical prelude from Tim Button's OpenSet Theory text.\end{editorial}\olimport{infinitesimals}\olimport{limits}\olimport{pathologies}\olimport{mythology}\olimport{cantor-plane}\olimport{hilbert-curve}\end{document}

content/history/set-theory/infinitesimals.tex

\documentclass[../../../include/open-logic-section]{subfiles}\begin{document}\olfileid{his}{set}{infinitesimals}\olsection{Infinitesimals and Differentiation}Newton and Leibniz discovered the calculus (independently) at the endof the 17th century. A particularly important application of thecalculus was \emph{differentiation}. Roughly speaking, differentiationaims to give a notion of the ``rate of change'', or gradient, of afunction at a point. Here is a vivid way to illustrate the idea.  Consider the function$f(x) = \nicefrac{x^2}{4} + \nicefrac{1}{2}$, depicted in black below:\begin{center}	\begin{tikzpicture}[scale=1]	\draw[->, gray] (-1,0) -- (4.25,0) node[right] {$x$};	\draw[->, gray] (0,-0.25) -- (0,5.25) node[above] {$f(x)$};		\foreach \x/\xtext in {1/1, 2/2, 3/3, 4/4}	\draw[shift={(\x,0)}, gray] (0pt,2pt) -- (0pt,-2pt) node[below] {$\xtext$};		\foreach \y/\ytext in {1/1, 2/2, 3/3, 4/4, 5/5}	\draw[shift={(0,\y)}, gray] (2pt,0pt) -- (-2pt,0pt) node[left] {$\ytext$};		\draw[oldiagcolorC] (0.5, 9/16) -- (3.5, 57/16) -- (3.5, 9/16)--cycle;	\draw[oldiagcolorD] (.5, 9/16) -- (2.5, 33/16) -- (2.5, 9/16) -- cycle;	\draw[oldiagcolorE] (.5, 9/16) -- (1.5, 17/16) -- (1.5, 9/16) -- cycle;	\draw[thick] (-1,.75) parabola bend (0,0.5) (4,4.5);	\end{tikzpicture}\end{center}Suppose we want to find the gradient of the function at $c =\nicefrac{1}{2}$. We start by drawing a triangle whose hypotenuseapproximates the gradient at that point, perhaps the red triangleabove. When $\beta$ is the base length of our triangle, its height is$f(\nicefrac{1}{2}+\beta) - f(\nicefrac{1}{2})$, so that the gradientof the hypotenuse is:\[\frac{f(\nicefrac{1}{2}+\beta) - f(\nicefrac{1}{2})}{\beta}.\]So the gradient of our !!{colorC} triangle, with base length~$3$, isexactly~$1$. The hypotenuse of a smaller triangle, the !!{colorD}triangle with base length~$2$, gives a better approximation; itsgradient is $\nicefrac{3}{4}$. A yet smaller triangle, the !!{colorE}triangle with base length~$1$, gives a yet better approximation; withgradient $\nicefrac{1}{2}$. Ever-smaller triangles give us ever-better approximations. So we mightsay something like this: the hypotenuse of a triangle with an\emph{infinitesimal} base length gives us the gradient at $c =\nicefrac{1}{2}$ itself. In this way, we would obtain a formula forthe (first) derivative of the function $f$ at the point $c$:\[{f'}(c) = \frac{f(c+\beta) - f(c)}{\beta} \text{ where $\beta$ is infinitesimal.}\]And, roughly, this is what Newton and Leibniz said. However, since they have said this, we must ask them: what is an\emph{infinitesimal}? A serious dilemma arises. If $\beta = 0$, then$f'$ is ill-defined, for it involves dividing by $0$. But if $\beta >0$, then we just get an \emph{approximation} to the gradient, and notthe gradient itself. This is not an anachronistic concern. Here is Berkeley, criticizingNewton's followers:\begin{quote}I admit that signs may be made to denote either any thing or nothing:and consequently that in the original notation $c + \beta$, $\beta$might have signified either an increment or nothing. But then which ofthese soever you make it signify, you must argue consistently withsuch its signification, and not proceed upon a double meaning: Whichto do were a manifest sophism. (\citealt[\S{}XIII]{Berkeley1734},variables changed to match preceding text)\end{quote}To defend the infinitesimal calculus against Berkeley, one might replythat the talk of ``infinitesimals'' is merely figurative. One mightsay that, so long as we take a \emph{really small} triangle, we willget a \emph{good enough} approximation to the tangent. Berkeley had areply to this too: whilst that might be good enough for engineering,it undermines the \emph{status} of mathematics,  for\begin{quote}we are told that \emph{in rebus mathematicis errores qu\`{a}m miniminon sunt contemnendi}. [In the case of mathematics, the smallesterrors are not to be neglected.] \citep[\S{}IX]{Berkeley1734}\end{quote}The italicised passage is a near-verbatim quote from Newton's own\emph{Quadrature of Curves} (1704). Berkeley's philosophical objections are deeply incisive. Nevertheless,the calculus was a massively successful enterprise, and mathematicianscontinued to use it without falling into error.\end{document}

content/history/set-theory/limits.tex

% Part: history% Chapter: set-theory% Section: limits\documentclass[../../../include/open-logic-section]{subfiles}\begin{document}	\olfileid{his}{set}{limits}	\olsection{Rigorous Definition of Limits}	These days, the standard solution to the foregoing problem is to getrid of the infinitesimals. Here is how. We saw that, as $\beta$ gets smaller, we get better approximations ofthe gradient. Indeed, as $\beta$ gets arbitrarily close to $0$, thevalue of $f'(c)$ ``tends without limit'' to the gradient we want. So,instead of considering what happens \emph{at} $\beta = 0$, we needonly consider the \emph{trend} of $f'(c)$ as $\beta$ approaches $0$. Put like this, the general challenge is to make sense of claims ofthis shape:\begin{center}As $x$ approaches $c$, $g(x)$ tends without limit to $\ell$. \end{center}which we can write more compactly as follows:\[\lim_{x \rightarrow c}g(x) = \ell.\]In the 19th century, building upon earlier work by Cauchy, Weierstrassoffered a perfectly rigorous definition of this expression. The ideais indeed that we can make $g(x)$ as close as we like to~$\ell$, bymaking $x$ suitably close to~$c$. More precisely, we stipulate that$\lim_{x \rightarrow c} g(x) = \ell$ will mean:\[(\forall\epsilon > 0)(\exists \delta > 0)\forall x \left(|x - c| < \delta \lif |g(x) - \ell| < \epsilon \right).\]The vertical bars here indicate absolute magnitude. That is, $|x| = x$when $x \geq 0$, and $|x| =-x$ when $x < 0$; you can depict\emph{that} function as follows:\begin{center}	\begin{tikzpicture}[scale=1]	\draw[->, gray] (-2.5,0) -- (2.5,0) node[right] {$x$};	\draw[->, gray] (0,-.2) -- (0,2.5) node[above] {$|x|$};		\foreach \x/\xtext in {-2/-2, -1,1, 1/1, 2/2}	\draw[shift={(\x,0)}] (0pt,2pt) -- (0pt,-2pt) node[below] {{$\xtext$}};		\foreach \y/\ytext in {1/1, 2/2}	\draw[shift={(0,\y)}] (2pt,0pt) -- (-2pt,0pt) node[left] {{$\ytext$}};		\draw[thick] (-2.5, 2.5)--(0,0)--(2.5, 2.5);	\end{tikzpicture}\end{center}So the definition says roughly this: you can make your ``error'' lessthan $\epsilon$ (i.e., $|g(x) - \ell| < \epsilon$) by choosingarguments which are no more than $\delta$ away from~$c$ (i.e., $|x -c| < \delta$). Having defined the notion of a limit, we can use it to avoidinfinitesimals altogether, stipulating that the gradient of $f$ at $c$is given by:\[	{f}'(c) = \lim_{x \rightarrow 0}\left(\frac{f(c +x) - f(c)}{x}\right) \text{ where a limit exists}.\]It is important, though, to realise why our definition needs thecaveat ``where a limit exists''. To take a simple example, consider$f(x) = |x|$, whose graph we just saw. Evidently, $f'(0)$ isill-defined: if we approach $0$ ``from the right'', the gradient isalways $1$; if we approach $0$ ``from the left'', the gradient isalways $-1$; so the limit is undefined. As such, we might add that afunction~$f$ is \emph{differentiable} at~$x$ iff such a limit exists.We have seen how to handle differentiation using the notion of a\emph{limit}. We can use the same notion to define the idea of a\emph{continuous} function. (Bolzano had, in effect, realised this by1817.) The Cauchy--Weierstrass treatment of continuity is as follows.Roughly: a function~$f$ is continuous (at a point) provided that, ifyou demand a certain amount of precision concerning the output of thefunction, you can guarantee this by insisting upon a certain amount ofprecision concerning the input of the function. More precisely: $f$ iscontinuous at $c$ provided that, as $x$ tends to zero, the differencebetween $f(c + x)$ and $f(c)$ itself tends to $0$. Otherwise put: $f$is \emph{continuous} at $c$ iff $f(c) = \lim_{x \rightarrow c} f(x)$. To go any further would just lead us off into real analysis, when oursubject matter is set theory. So now we should pause, and state themoral. During the 19th century, mathematicians learnt how to dowithout infinitesimals, by invoking a rigorously defined notion of a\emph{limit}.\end{document}

content/history/set-theory/pathologies.tex

\documentclass[../../../include/open-logic-section]{subfiles}\begin{document}\olfileid{his}{set}{pathology}\olsection{Pathologies}However, the definition of a \emph{limit} turned out to allow for some rather ``pathological'' constructions. Around the 1830s, Bolzano discovered a function which was\emph{continuous everywhere}, but \emph{differentiable nowhere}.(Unfortunately, Bolzano never published this; the idea was firstencountered by mathematicians in 1872, thanks to Weierstrass'sindependent discovery of the same idea.)\footnote{The history isdocumented in extremely thorough footnotes to the Wikipedia article on\href{http://en.wikipedia.org/wiki/Weierstrass_function}{theWeierstrass function}.} This was, to say the least, rather surprising.It is easy to find functions, such as $|x|$, which are continuouseverywhere but not differentiable at a particular point. But afunction which is continuous everywhere but differentiable\emph{nowhere} is a very different beast. Consider, for a moment, howyou might try to draw such a function. To ensure it is continuous, youmust be able to draw it without ever removing your pen from the page;but to ensure it is differentiable nowhere, you would have to abruptlychange the direction of your pen, constantly.Further ``pathologies'' followed. In January 5 1874, Cantor wrote aletter to Dedekind, posing the problem:\begin{quote}Can a surface (say a square including its boundary) be one-to-onecorrelated to a line (say a straight line including its endpoints) sothat to every point of the surface there corresponds a point of theline, and conversely to every point of the line there corresponds apoint of the surface?It still seems to me at the moment that the answer to this question isvery difficult---although here too one is so impelled to say \emph{no}that one would like to hold the proof to be almost superfluous.[Quoted in \citealt{Gouvea2011}]\end{quote}But, in 1877, Cantor proved that he had been wrong. In fact, a lineand a square have exactly the same number of points. He wrote on 29June 1877 to Dedekind ``\emph{je le vois, mais je ne le crois pas}'';that is, ``I see it, but I don't believe it''. In the ``receivedhistory'' of mathematics, this is often taken to indicate just how\emph{literally incredible} these new results were to themathematicians of the time. (The correspondence is presented in\citet{Gouvea2011}, and we return to it in\olref[his][set][mythology]{sec}. Cantor's proof is outlined in\olref[his][set][cantorplane]{sec}.) Inspired by Cantor's result, Peano started to consider whether itmight be possible to map a line \emph{smoothly} onto a plane. Thiswould be a \emph{curve which fills space}. In \citeyear{Peano1890},Peano constructed just such a curve. This is truly counter-intuitive:Euclid had defined a line as ``breadthless length'' (Book I,Definition 2), but Peano had shown that, by curling up a lineappropriately, its length can be turned into breadth. In\citeyear{Hilbert1891}, Hilbert described a slightly more intuitivespace-filling curve, together with some pictures illustrating it. Thecurve is constructed in sequence, and here are the first six stages ofthe construction:\begin{center}\begin{tikzpicture}\begin{scope}[xshift=20pt, yshift=20pt]	\draw [oldiagcolorC, l-system={Hilbert curve, step=40pt, angle=90, axiom=L, order=1}] lindenmayer system;\end{scope}\begin{scope}[xshift=110pt, yshift=10pt]	\draw [oldiagcolorC, l-system={Hilbert curve, step=20pt, angle=90, axiom=L, order=2}] lindenmayer system;\end{scope}\begin{scope}[xshift=205pt, yshift=5pt]	\draw [oldiagcolorC, l-system={Hilbert curve, step=10pt, angle=90, axiom=L, order=3}] lindenmayer system;\end{scope}\begin{scope}[xshift=2.5pt, yshift=-97.5pt]	\draw [oldiagcolorC, l-system={Hilbert curve, step=5pt, angle=90, axiom=L, order=4}] lindenmayer system;\end{scope}\begin{scope}[xshift=101.25pt, yshift=-98.75pt]	\draw [oldiagcolorC, l-system={Hilbert curve, step=2.5pt, angle=90, axiom=L, order=5}] lindenmayer system;\end{scope}\begin{scope}[xshift=200.625pt, yshift=-99.375pt]	\draw [oldiagcolorC, l-system={Hilbert curve, step=1.25pt, angle=90, axiom=L, order=6}] lindenmayer system;\end{scope}\draw[gray] (0pt,0pt) rectangle (80pt, 80pt);\draw[gray] (100pt,0pt) rectangle (180pt, 80pt);\draw[gray] (200pt,0pt) rectangle (280pt, 80pt);\draw[gray] (0pt,-100pt) rectangle (80pt, -20pt);\draw[gray] (100pt,-100pt) rectangle (180pt, -20pt);\draw[gray] (200pt,-100pt) rectangle (280pt, -20pt);\end{tikzpicture}  \end{center}In the limit---a notion which had, by now, received rigorousdefinition---the entire square is filled in solid !!{colorC}. And, inpassing, Hilbert's curve is continuous everywhere but differentiablenowhere; intuitively because, in the infinite limit, the functionabruptly changes direction at every moment. (We will outline Hilbert'sconstruction in more detail in \olref[his][set][hilbertcurve]{sec}.)For better or worse, these ``pathological'' geometric constructionswere treated as a reason to doubt appeals to geometric intuition. Theybecame something approaching \emph{propaganda} for a new way of doingmathematics, which would culminate in set theory. In the latermyth-building of the subject, it was repeated, often, that theseresults were both perfectly rigorous and perfectly shocking. Theytherefore served a dual purpose: as a warning against relying upongeometric intuition, and as a demonstration of the fertility of newways of thinking. \end{document}

content/history/set-theory/mythology.tex

\documentclass[../../../include/open-logic-section]{subfiles}\begin{document}\olfileid{his}{set}{mythology}\olsection{More Myth than History?}Looking back on these events with more than a century of hindsight, wemust be careful not to take these verdicts on trust. The results werecertainly novel, exciting, and surprising. But how truly shocking werethey? And did they really demonstrate that we should not rely ongeometric intuition?On the question of shock, \citet{Gouvea2011} points out that Cantor'sfamous note to Dedekind, ``\emph{je le vois, mais je ne le croispas}'' is taken rather out of context. Here is more of that context(quoted from \citeauthor{Gouvea2011}):\begin{quote}Please excuse my zeal for the subject if I make so many demands uponyour kindness and patience; the communications which I lately sent youare even for me so unexpected, so new, that I can have no peace ofmind until I obtain from you, honoured friend, a decision about theircorrectness. So long as you have not agreed with me, I can only say:\emph{je le vois, mais je ne le crois pas.} \end{quote}Cantor knew his result was ``so unexpected, so new''. But it isdoubtful that he ever found his result \emph{unbelievable}. As\citeauthor{Gouvea2011} points out, he was simply asking Dedekind tocheck the proof he had offered. On the question of geometric intuition: Peano published hisspace-filling curve without including any diagrams. But when Hilbertpublished his curve, he explained his purpose: he would providereaders with a clear way to understand Peano's result, if they ``helpthemselves to the following geometric intuition''; whereupon heincluded a series of \emph{diagrams} just like those provided in\olref[his][set][pathology]{sec}. More generally: whilst diagrams have fallen rather out of fashion inpublished proofs, there is no getting round the fact thatmathematicians \emph{frequently} use diagrams when proving things.(Roughly put: good mathematicians know when they can rely upongeometric intuition.)In short: don't believe the hype; or at least, don't just take it ontrust. For more on this, you could read \citet{Giaquinto2007}.\end{document}

content/history/set-theory/cantor-plane.tex

\documentclass[../../../include/open-logic-section]{subfiles}\begin{document}\olfileid{his}{set}{cantorplane}\olsection{Cantor on the Line and the Plane}Some of the circumstances surrounding the proof ofSchr\"oder-Bernstein tie in with the history we discussed in\olref[his][set][pathology]{sec}. Recall that, in 1877,Cantor proved that there are exactly as many points on a square as onone of its sides. Here, we will present his (first attempted) proof.Let $\unitline$ be the unit line, i.e., the set of points $[0,1]$. Let$\unitsquare$ be the unit square, i.e., the set of points $\unitline\times \unitline$. In these terms, Cantor proved that$\cardeq{\unitline}{\unitsquare}$. He wrote a note to Dedekind,essentially containing the following argument.\begin{thm}\ollabel{thm:cantorplane}$\cardeq{\unitline}{\unitsquare}$\end{thm}\begin{proof}[Proof: first part.] Fix $a, b \in \unitline$. Write them in binary notation, so that wehave infinite sequences of $0$s and $1$s, $a_1$, $a_2$, \dots, and$b_1$, $b_2$, \dots, such that:\begin{align*}a &= 0.a_1a_2a_3a_4\dots\\b &= 0.b_1b_2b_3b_4\dots\intertext{Now consider the function $f \colon \unitsquare \to \unitline$ given by} f(a, b) & = 0.a_1b_1a_2b_2a_3b_3a_4b_4\dots\end{align*}Now $f$ is !!a{injection}, since if $f(a, b) = f(c,d)$, then  $a_n =c_n$ and $b_n = d_n$ for all $n \in \Nat$, so that $a = c$ and $b =d$.\end{proof}Unfortunately, as Dedekind pointed out to Cantor, this does not answerthe original question. Consider $0.\dot{1}\dot{0} =0.1010101010\ldots$. We need that $f(a,b) = 0.\dot{1}\dot{0}$, where:\begin{align*}a&= 0.\dot{1}\dot{1} = 0.111111\ldots\\b&= 0\end{align*}But $a = 0.\dot{1}\dot{1} = 1$. So, when we say ``write $a$ and $b$ inbinary notation'', we have to choose \emph{which} notation to use;and, since $f$ is to be a \emph{function}, we can use only \emph{one}of the two possible notations. But if, for example, we use the simplenotation, and write $a$ as ``$1.000\ldots$'', then we have no pair$\tuple{a, b}$ such that $f(a, b) = 0.\dot{1}\dot{0}$. To summarise: Dedekind pointed out that, given the possibility ofcertain recurring decimal expansions, Cantor's function $f$ is!!a{injection} but \emph{not} !!a{surjection}. So Cantor has shownonly that $\cardle{\unitsquare}{\unitline}$ and \emph{not} that$\cardeq{\unitsquare}{\unitline}$. Cantor wrote back to Dedekind almost immediately, essentiallysuggesting that the proof could be completed as follows:\begin{proof}[Proof: completed.] So, we have shown that $\cardle{\unitsquare}{\unitline}$. But there isobviously !!a{injection} from $\unitline$ to $\unitsquare$: just laythe line flat along one side of the square. So$\cardle{\unitline}{\unitsquare}$ and$\cardle{\unitsquare}{\unitline}$. By Schr\"{o}der--Bernstein(\olref[sfr][siz][sb]{thm:schroder-bernstein}),$\cardeq{\unitline}{\unitsquare}$.\end{proof}But of course, Cantor could not complete the last line in these terms,for the Schr\"{o}der-Bernstein Theorem was not yet proved. Indeed,although Cantor would subsequently formulate this as a generalconjecture, it was not satisfactorily proved until 1897. (And so,later in 1877, Cantor offered a different proof of\olref{thm:cantorplane}, which did not go viaSchr\"{o}der--Bernstein.)\end{document}

content/history/set-theory/hilbert-curve.tex

\documentclass[../../../include/open-logic-section]{subfiles}\begin{document}\olfileid{his}{set}{hilbertcurve}\olsection{Appendix: Hilbert's Space-filling Curves}In chapter \olref[pathology]{sec}, we mentioned that Cantor's proof thata line and a square have exactly the same number of points(\olref[his][set][cantorplane]{thm:cantorplane}) prompted Peano toask whether there might be a space-filling \emph{curve}. He obtained apositive answer in 1890. In this section, we explain (in ahand-wavy way) how to construct Hilbert's space-filling curve (with atiny tweak).\footnote{For a more rigorous explanation, see\cite{Rose2010}. The tweak amounts to the inclusion of the redparts of the curves below. This makes it slightly easier to check thatthe curve is continuous.}We must define a function, $h$, as the limit of a sequence of functions $h_1$, $h_2$, $h_3$, \dots\@ We first describe the construction. Then we show it is space-filling. Then we show it is a curve. We will take $h$'s range to be the unit square, $\unitsquare$. Here is our first approximation to $h$, i.e., $h_1$:\begin{center}	\begin{tikzpicture}	\draw[gray] (0pt, 0pt) rectangle (80pt, 80pt);	\draw[step = 40pt, gray, very thin] (0pt, 0pt) grid (80pt, 80pt);	\draw[oldiagcolorC, thick] (20pt, 0)--(20pt, 20pt);	\draw[oldiagcolorC, thick] (60pt, 0)--(60pt, 20pt);			\begin{scope}[xshift=20pt, yshift=20pt]	\draw [black, thick, l-system={Hilbert curve, step=40pt, angle=90, axiom=L, order=1}]  lindenmayer system;	\end{scope}	\end{tikzpicture}\end{center}To keep track of things, we have imposed a $2 \times 2$ grid on the square. We can think of the curve starting in the bottom left quarter, moving to the top left, then to the top right, then finally to the bottom right. Here is the second stage in the construction, i.e., $h_2$:\begin{center}	\begin{tikzpicture}	\draw[gray] (0pt, 0pt) rectangle (80pt, 80pt);	\draw[step = 20pt, gray, very thin] (0pt, 0pt) grid (80pt, 80pt);	\draw[oldiagcolorC, thick] (0pt, 10pt)--(10pt, 10pt);	\draw[oldiagcolorC, thick] (70pt, 10pt)--(80pt, 10pt);	\draw[thick, oldiagcolorE] (10pt, 30pt)--(10pt, 50pt); 	\draw[thick, oldiagcolorE] (30pt, 50pt)--(50pt, 50pt); 	\draw[thick, oldiagcolorE] (70pt, 50pt)--(70pt, 30pt); \begin{scope}[xshift=10pt, yshift=30pt]	\draw [thick, rotate=270,black, l-system={Hilbert curve, step=20pt, angle=90, axiom=L, order=1}]  lindenmayer system;	\end{scope}	\begin{scope}[xshift=10pt, yshift=50pt]	\draw [thick, black, l-system={Hilbert curve, step=20pt, angle=90, axiom=L, order=1}]  lindenmayer system;	\end{scope}	\begin{scope}[xshift=50pt, yshift=50pt]	\draw [thick, black, l-system={Hilbert curve, step=20pt, angle=90, axiom=L, order=1}]  lindenmayer system;	\end{scope}	\begin{scope}[xshift=70pt, yshift=10pt]	\draw [thick, rotate=90,black, l-system={Hilbert curve, step=20pt, angle=90, axiom=L, order=1}]  lindenmayer system;	\end{scope}	\end{tikzpicture}\end{center}The different colours will help explain how $h_2$ was constructed. We first place scaled-down copies of the non-!!{colorC} bit of $h_1$ into the bottom left, top left, top right, and bottom right of our square (drawn in black). We then connect these four figures (with !!{colorE} lines). Finally, we connect our figure to the boundary of the square (with !!{colorC} lines).Now to $h_3$. Just as $h_2$ was made from four connected, scaled-down copies of the non-red bit of $h_1$, so $h_3$ is made up of four scaled-down copies of the non-red bit of $h_2$ (drawn in black), which are then joined together (with !!{colorE} lines) and finally connected to the boundary of the square (with !!{colorC} lines).\begin{center}	\begin{tikzpicture}	\draw[gray] (0pt, 0pt) rectangle (80pt, 80pt);	\draw[step = 10pt, gray, very thin] (0pt, 0pt) grid (80pt, 80pt);	\draw[thick, oldiagcolorC](5pt, 0pt)--(5pt, 5pt); 	\draw[thick, oldiagcolorC] (75pt, 0pt)--(75pt, 5pt); 	\draw[thick, oldiagcolorE] (75pt, 45pt)--(75pt, 35pt);	\draw[thick, oldiagcolorE] (5pt, 35pt)--(5pt, 45pt); 	\draw[thick, oldiagcolorE] (35pt, 45pt)--(45pt, 45pt); 	\draw[thick, oldiagcolorE] (75pt, 45pt)--(75pt, 35pt); \begin{scope}[xshift=5pt, yshift=35pt]	\draw [rotate=270, thick, black, l-system={Hilbert curve, step=10pt, angle=90, axiom=L, order=2}]  lindenmayer system;	\end{scope}	\begin{scope}[xshift=5pt, yshift=45pt]	\draw [thick, black, l-system={Hilbert curve, step=10pt, angle=90, axiom=L, order=2}]  lindenmayer system;	\end{scope}	\begin{scope}[xshift=45pt, yshift=45pt]	\draw [thick, black, l-system={Hilbert curve, step=10pt, angle=90, axiom=L, order=2}]  lindenmayer system;	\end{scope}	\begin{scope}[xshift=75pt, yshift=5pt]	\draw [thick, rotate=90,black, l-system={Hilbert curve, step=10pt, angle=90, axiom=L, order=2}]  lindenmayer system;	\end{scope}	\end{tikzpicture}\end{center}And now we see the general pattern for defining $h_{n+1}$ from $h_n$.At last we define the curve $h$ \emph{itself} by considering thepoint-by-point limit of these successive functions $h_1$, $h_2$,\dots\@ That is, for each $x \in \unitsquare$:\begin{align*}	h(x) &= \lim_{n \rightarrow \infty} h_n(x)\end{align*} We now show that this curve fills space. When we draw the curve $h_n$,we impose a $2^n \times 2^n$ grid onto $\unitsquare$. By Pythagoras'sTheorem, the diagonal of each grid-location is of length:\[\sqrt{\left(\nicefrac{1}{2^{n}}\right)^2+\left(\nicefrac{1}{2^{n}}\right)^2} = 2^{(\frac{1}{2}-n)}\]and evidently $h_n$ passes through every grid-location. So each pointin $\unitsquare$ is \emph{at most} $2^{(\frac{1}{2}-n)}$ distance awayfrom some point on $h_n$. Now, $h$ is defined as the limit of thefunctions $h_1$, $h_2$, $h_3$, \dots\@ So the maximum distance of anypoint from $h$ is given by:\[\lim_{n \rightarrow \infty} 2^{(\frac{1}{2}-n)} = 0.\]That is: every point in $\unitsquare$ is $0$ distance from~$h$. Inother words, every point of $\unitsquare$ lies \emph{on} the curve. So $h$fills space!{}It remains to show that $h$ is, indeed, a \emph{curve}. To show this,we must define the notion. The modern definition builds on one givenby Jordan in 1887 (i.e., only a few years before the firstspace-filling curve was provided): \begin{defn}A curve is a continuous map from $\unitline$ to $\Real^2$. \end{defn}This is fairly intuitive: a curve is, intuitively, a ``smooth'' mapwhich takes a canonical line onto the plane $\Real^2$. Our function,$h$, is indeed a map from $\unitline$ to $\Real^2$. So, we just needto show that $h$ is continuous. We defined continuity in\olref[limits]{sec} using $\epsilon$/$\delta$ notation. In thevernacular, we want to establish the following: \emph{If you specify apoint $p$ in $\unitsquare$, together with any desired level ofprecision $\epsilon$, we can find an open section of $\unitline$ suchthat, given any $x$ in that open section, $h(x)$ is within $\epsilon$of $p$.}So: assume that you have specified $p$ and $\epsilon$. This is, ineffect, to draw a circle with centre $p$ and radius $\epsilon$ on$\unitsquare$. (The circle might spill off the edge of $\unitsquare$,but that doesn't matter.) Now, recall that, when describing thefunction $h_n$, we drew a $2^n \times 2^n$ grid upon $\unitsquare$. Itis obvious that, no matter how small $\epsilon$ is, there is some $n$such that some individual grid-location of the $2^n \times 2^n$ gridon $\unitsquare$ lies wholly within the circle with centre $p$ andradius $\epsilon$. So, take that $n$, and let $I$ be the largest open part of $\unitline$which $h_n$ maps wholly into the relevant grid location. (It is clearthat $(a,b)$ exists, since we already noted that $h_n$ passes throughevery grid-location in the $2^n\times 2^n$ grid.) It now suffices toshow to show that, whenever $x \in I$ the point $h(x)$ lies in thatsame grid-location. And to do \emph{this}, it suffices to show that$h_m(x)$ lies in that same grid location, for any $m > n$. But this isobvious. If we consider what happens with $h_m$ for $m > n$, we seethat exactly the ``same part'' of the unit interval is mappedinto the same grid-location; we just map it into that region in anincreasingly stretched-out, wiggly fashion. \end{document}