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To create a pdf file from this %use pdflatex % % if you read one more comment read the one that starts with *********** % % %\input{/Users/dwick/Library/texmf/tex/latex/misc/defs2} %See folder texpower for more GREAT EXAMPLES %LIKE THIS. \nofiles % Suppress all those pesky files! \include{defs2} \def\bi{\begin{itemize}} \def\ei{\end{itemize}} \def\ob{\overline} \def\pii{\noi \item} %\def\noibb {\noi \hspace{.5in} \vbox{$\circ \ $ }} \renewcommand{\labelitemii}{$\circ$} \def\next{\vfill\eject\noindent} % include some color support (see examples below) \usepackage[usenames,dvipsnames]{color} %Include the texpower packages to allow for pause \usepackage[display]{texpower} \usepackage{ fancyvrb,listings,alltt} %\usepackage{pdfslide} \newcommand{\tabend}{\\\hline} % % set background color \definecolor{Pcolor}{rgb}{.971,.951,.92} % A cool manila \pagecolor{Pcolor} % see the Tex files associated with dvipsnames % for some predefined colors. % % this little style file allows you to % step through a page adding items % %\usepackage{pause} % % include a graphics package for importing figures % \usepackage{graphicx} % % \RequirePackage[pdftex,pdfpagemode=none, pdftoolbar=true, pdffitwindow=true,pdfcenterwindow=true]{hyperref} % % change these to suit! Margins assume an inch as default % \setlength{\paperheight}{7in} \setlength{\paperwidth}{9in} \setlength{\textheight}{6.75in} \setlength{\textwidth}{7.5in} \setlength{\evensidemargin}{-.75in} \setlength{\oddsidemargin}{-.75in} \setlength{\topmargin}{-.75in} \setlength{\parskip}{.125in} \setlength{\parindent}{0in} % % % set background color \definecolor{Pcolor}{rgb}{.971,.951,.92} % A rosy manila \pagecolor{Pcolor} % Define heading color. \definecolor{Hcolor}{rgb}{.0,.0,.99} % my heading color % % these are my basic page headings % note how the color is set % \def\mytitle#1{{\color{Hcolor}{\it \LARGE #1}\\ }} \def\MYTITLE#1{{\vspace{-.5in}\bf\color{Hcolor}{\LARGE #1}\\ \rule[.15in]{\textwidth}{.03in} } \\ \vspace{-.7in}\noindent} \def\myTITLE#1{{\color{Hcolor}{\huge #1}\\}} % just to make the math work, everyone does this differently ... \newcommand{\BLD}[1]{\mbox{\boldmath $#1$}} % low tech way to start a new "slide" and add a background image % This has been tuned to the aspect paper sizes declared above. \newcommand{\BS}{ \newpage \vspace*{-1.26in} \hspace*{-.32in} \includegraphics[height=7.2in, width=\paperwidth]{/home/dwick/DATA/Professional/Other/Pictures&Worksheets/17a.jpg} % Some rock face in Boulder Canyon. Time-For-Lunch-2.jpg \vspace*{-7.3in} \vspace*{.5in} \hspace*{-\paperwidth} } % uncomment this version for just a new page and no background image. % note that you still get a background color from setting \pagecolor above %\newcommand{\BS}{ \newpage} \begin{document} %%%%%%%%%%% {\color{Black} %begin default text color {\LARGE % fault is LARGE size font { \sf % default face is sans serif %%%%%%%%%%%% \BS \cl {Georg Cantor's Infinities,} \cl {Paradoxes,} \cl {and the} \cl {Foundations of Mathematics} %%%%%%%%%%%% \BS \MYTITLE {GEORG CANTOR (1845-1918)} \noi \cl {\includegraphics[width=4in,height=3.5in]{/home/dwick/DATA/Professional/Other/Math/math_photos/Cantor.jpg}} \cl{Father of Set Theory} \BS \MYTITLE{What is a Set?} \noi \cl {Cantor: \it "A set is a collection, thought of as a whole"} \BS \MYTITLE {Examples of Sets} \noi \noi $$A=\{1,2,3\}$$ \noi \noi $$B=\{my \ immediate \ family \}$$ \noi \noi $$\fatn =\{1,2,3,\cdots\}$$ \BS \noi \cl {Can compare size (i.e. {\it cardinality}) of finite sets} \cl{ by {\it pairing} elements} \noi $$|\{1,2,3\}|=|\{a,b,c\}|=3$$ \noi $$1\leftrightarrow a $$\noi$$ 2\leftrightarrow c $$\noi$$ 3\leftrightarrow b$$ \noi \BS $$3=|\{a,b,c\}| < |\{ my \ immediate \ family \} |=5$$ \noi $$a\leftrightarrow Julie $$\noi$$ b\leftrightarrow Tanner$$\noi$$ c\leftrightarrow Tatum$$ \noi \noi \cl {STUFF LEFT OVER} \BS Why not extrapolate this idea to infinite sets????? \noi {\bf Previous to Cantor:} Mathematicians had been somewhat leery of the notion of the infinite, and did their best to avoid it. \noi \noi Aristotle, in dealing with Zeno's paradoxes, concluded that the way to resolve them was to deny the possibility of collecting infinitely many objects into a complete and actually existing whole. \noi \noi \cl { {\it Potential} $\infty$ vs. {\it Completed} $\infty$} \BS \MYTITLE { GALILEO GALILEI (1564-1642)} \noi \cl{\includegraphics[]{/home/dwick/DATA/Professional/Other/Math/math_photos/Galileo.jpg}} \BS \MYTITLE { GALILEO GALILEI (1564-1642)} \cl {Noticed the following pairing} \noi $$1\leftrightarrow 1$$\noi$$ 2\leftrightarrow 4 $$\noi$$ 3\leftrightarrow 9 $$\noi$$4\leftrightarrow 16 $$\noi $$5\leftrightarrow 25 $$ $$\cdots \cdots$$ \BS This describes a pairing between the set $\fatn$ of all natural numbers, \noi and the set $\fatn^2$ of all perfect squares. \noi \noi So there seems to be the {\it same} number of perfect squares as there are natural numbers. \noi Although in another sense there seem to be less perfect squares than natural numbers. \noi \noi Galileo's Response: \noi \noi \cl {YIKES!} \BS \MYTITLE {BERNARD BOLZANO (1781-1848)} \noi \cl{\includegraphics[]{/home/dwick/DATA/Professional/Other/Math/math_photos/Bolzano_3.jpg}} \BS \MYTITLE {BERNARD BOLZANO (1781-1848)} \noi The intervals $[0,1]$ and $[0,2]$ have the same cardinality via the line $y=2x$ \noi \noi \hspace{2.5in} \includegraphics[height=2in,width=3in]{/home/dwick/DATA/Professional/Other/Math/math_photos/bolzano.pdf} \noi Although one of the intervals is twice as long, they have the same cardinality. \BS Bolzano's ideas received little attention \noi Previous to Cantor: "We cannot concieve of infinitely many things" \noi \noi Cantor's Response: \noi \noi \cl {"Phooey"} \BS \MYTITLE {CANTOR (again)} \noi "Let us complete infinity" \noi \noi $$\fatn=\{ 1,2,3,\ldots \}$$ \noi $$|\fatn |=|\fatn^2|=\aleph_0 = \ smallest \ infinity$$ \noi \noi Any set X with cardinality $\aleph_0$ is called \ub {countable} (i.e. there exists a pairing between $\fatn$ and X) \BS Question: What other sets are countable?? \noi \noi \cl {$\fatz =\{ \cdots ,-3,-2,-1,0,1,2,3, \cdots \} $} \noi \noi $$1\leftrightarrow 0$$$$ 2\leftrightarrow 1 $$$$ 3\leftrightarrow -1 $$$$4\leftrightarrow 2 $$ $$5\leftrightarrow -2 $$$$ 6\leftrightarrow 3 $$$$ 7\leftrightarrow -3 $$$$ 8\leftrightarrow 4 $$$$\cdots$$ \BS In other words, to show that a set is countable, we only need to display an ordering where there is a first, second, third, fourth, etc. \noi \noi \cl {$\fatz =\{ \cdots ,-3,-2,-1,0,1,2,3, \cdots \} $} \noi \noi \cl {$\fatz =\{ 0,1,-1,2,-2,3,-3,4,-4, \cdots \} $} \noi \noi $$|\ \fatz|=|\fatn|=\aleph_0$$ \BS \cl {$\fatq$ = set of all fractions } \noi \noi $${1\over 1} \ \ \ {2\over 1} \ \ \ {3\over 1} \ \ \ {4\over 1} \ \ \ {5\over 1} \ \ \ \cdots \cdots \cdots$$ \noi $${1\over 2} \ \ \ {2\over 2} \ \ \ {3\over 2} \ \ \ {4\over 2} \ \ \ {5\over 2} \ \ \ \cdots \cdots \cdots$$ \noi $${1\over 3} \ \ \ {2\over 3} \ \ \ {3\over 3} \ \ \ {4\over 3} \ \ \ {5\over 3} \ \ \ \cdots \cdots \cdots$$ \noi $${1\over 4} \ \ \ {2\over 4} \ \ \ {3\over 4} \ \ \ {4\over 4} \ \ \ {5\over 4} \ \ \ \cdots \cdots \cdots$$ \noi $${1\over 5} \ \ \ {2\over 5} \ \ \ {3\over 5} \ \ \ {4\over 5} \ \ \ {5\over 5} \ \ \ \cdots \cdots \cdots$$ $$\cdots \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$$ $$\cdots \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$$ \BS $$|\ \fatq|=\aleph_0$$ \BS Question: What sets are not countable? \noi \noi Let $(0,1) $ denote the interval of all {\it real} numbers between $0$ and $1$. \noi Note that this includes not only the fractions between $0$ and $1$ but also the {\it irrational} numbers like $\pi -3$ that cannot be expressed as fractions. \noi Is $(0,1)$ countable? \BS Suppose there exist a pairing between $\fatn$ and $ (0,1)$ \noi $$1\leftrightarrow.a_{11}a_{12}a_{13}a_{14}\cdots\cdots\cdots$$ $$2\leftrightarrow.a_{21}a_{22}a_{23}a_{24}\cdots\cdots\cdots$$ $$3\leftrightarrow.a_{31}a_{32}a_{33}a_{34}\cdots\cdots\cdots$$ $$4\leftrightarrow.a_{41}a_{42}a_{43}a_{44}\cdots\cdots\cdots$$ $$\cdots \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$$ \noi $${\rm Define} \ x=.c_1c_2c_3c_4..........$$ \cl {where $c_i =6 \ if \ a_{ii}\not= 6$ and } $$c_i=5 \ {\rm otherwise}$$ \BS For example, suppose our list of numbers was as below $$1\leftrightarrow.1234\cdots\cdots\cdots$$ $$2\leftrightarrow.0632\cdots\cdots\cdots$$ $$3\leftrightarrow.6575\cdots\cdots\cdots$$ $$\cdots \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$$ \noi $$ \ x=.c_1c_2c_3c_4..........$$ \cl {where $c_i =6 \ if \ a_{ii}\not= 6$ and } $$c_i=5 \ {\rm otherwise}$$ \noi $$x=.656\cdots$$ \BS Then $x$ differs from each number in the list in the ${ii}^{th}$ spot. $$1\leftrightarrow.{\bf a_{11}}a_{12}a_{13}a_{14}\cdots\cdots\cdots$$ $$2\leftrightarrow.a_{21}{\bf a_{22}}a_{23}a_{24}\cdots\cdots\cdots$$ $$3\leftrightarrow.a_{31}a_{32}{\bf a_{33}}a_{34}\cdots\cdots\cdots$$ $$4\leftrightarrow.a_{41}a_{42}a_{43}{\bf a_{44}}\cdots\cdots\cdots$$ $$\cdots \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$$ Thus the number $x$ is in the interval $(0,1)$ but is not paired with any natural number. \noi \noi $x$ is STUFF LEFT OVER! \noi $$|\fatr|> |\fatn|=\aleph_0$$ \noi \BS And by considering the power set of a set $A$ (i.e. the set of all subsets of the set $A$, denoted by $P(A)$ ) Cantor was able to establish by another devious argument that for any set $A$, \noi $$|P(A)|>|A|$$ \noi He thus established a hierchy of infinite cardinal numbers $$|\fatn | < |P({\fatn }) |< |P({P({\fatn }})) | < \cdots$$ \noi or $$\aleph_0 < \aleph_1 < \aleph_2 < \cdots $$ \noi where $$|\fatr|=\aleph_1$$ \BS Let me repeat this. It is important \noi Cantor established that there are infinitely many infinities: $$\aleph_0 < \aleph_1 < \aleph_2 < \cdots $$ \noi $$|\fatn|=| \ \fatz \ |=|\ \fatq|=\aleph_0$$ $$|\fatr|=\aleph_1$$ \BS \MYTITLE {CONTINUUM HYPOTHESIS} \noi CH: There is no set with cardinality between $\aleph_0$ and $\aleph_1$ \noi \noi Cantor was convinced the Continuum Hypothesis was true, and spent the rest of his life trying (unsuccesfully) to prove it. \noi \BS \MYTITLE {David Hilbert} \cl{ \includegraphics[height=4in,width=3in]{/home/dwick/DATA/Professional/Other/Math/math_photos/Hilbert.jpg}} \noi {\it "No one can expel us from the paradise Cantor has created"} \BS \MYTITLE {RUSSELL'S PARADOX} \noi $$F=\{all \ forks\}$$ \noi $$G=\{ all \ things \ that \ aren't \ a \ fork\}$$ \noi $$ G \ is \ not \ a \ fork$$ \noi $$F\notin F \ \ but \ \ G\in G$$ \noi \noi \cl {Some sets may contain themselves} \noi \noi \cl {HMMMMM?} \BS \MYTITLE {RUSSELL'S PARADOX} $$T=\{ sets \ that \ don't \ contain \ themselves\}$$ \noi \noi The elements of T are \underbar {all} of those sets that do not contain themselves: e.g. $$F\in T$$ $$G\notin T$$ \noi \noi Question: Is $T\in T$ ? \noi \BS \MYTITLE {RUSSELL'S PARADOX} Suppose $T\in T$. Then $T$ would contain itself. \noi Thus $T$ would not consist entirely of sets that don't contain themselves. \noi $$T\in T \Longrightarrow T\notin T$$ \BS \MYTITLE {RUSSELL'S PARADOX} Suppose $T\notin T$. Then $T$ doesn't contain itself. \noi Thus $T$ would not consist of {\it all} of the sets that don't contain themselves \noi (since $T$ is missing). \noi $$T\notin T \Longrightarrow T\in T$$ \noi \BS \MYTITLE {RUSSELL'S PARADOX} Conclusion? \noi Considering the set of all sets that don't contain themselves leads to logical contradictions. \noi \BS \MYTITLE {Henri Poincare} \cl {\includegraphics[height=4in,width=3in]{/home/dwick/DATA/Professional/Other/Math/math_photos/Poincare3.jpg}} \noi {\it Future generations of mathematicians will come to view Cantor's work as a disease from which we have recovered} \noi \BS \noindent Mathematicians tried to "fix" set theory by ridding it of the paradoxes and placing it (and ALL? of mathematics) on a firm foundation. \noi \noi How? By axiomatizing set theory (i.e. give precise and formal rules for constructing sets) \noi \noi Cantor=Naive Set Theory (his definition of set was too vague and allowed for unecessary nonsense) \noi \noi RULE 1: No set shall contain itself. \noi In the early $20^{th}$ century, mathematicians sought to axiomatize all areas of mathematics. \BS \MYTITLE{Zermelo/Fraenkel } \cl{\includegraphics[height=4in,width=3in]{/home/dwick/DATA/Professional/Other/Math/math_photos/Zermelo.jpg}} \noi Axioms of Set Theory \BS \MYTITLE{Hilbert/Bernays/Von Neumann} \cl{\includegraphics[height=4in,width=4in]{/home/dwick/DATA/Professional/Other/Math/math_photos/Von_Neumann.jpg}} \noi Set Theory and Geometry \noi \BS \MYTITLE{Peano/Frege} \cl{\includegraphics[height=4in,width=3in]{/home/dwick/DATA/Professional/Other/Math/math_photos/Frege.jpg}} \noi \noi Arithmetic \BS \MYTITLE{Russell/Whitehead} \cl{\includegraphics[height=4in,width=3.5in]{/home/dwick/DATA/Professional/Other/Math/math_photos/Russell.jpg}} \BS \MYTITLE{Russell/Whitehead} \noi {\it Principia Mathematica}: Reduce all of mathematics to logic. \noi \noi 250 pages to prove 1+1=2 \noi \noi Russell: "For 15 years I labored, and I finally came to the conclusion that there was nothing more I could do". \BS Zermelo was led to set theory by his investigation of the Well-Ordering Principle. \noi \noi Well-Ordering Principle (WOP): Any set can be ordered so that every non-empty subset has a least element. \noi \noi Examples: \noi \noi $$\fatn=\{ 1,2,3,\ldots\}$$ \noi \noi $$\fatz =\{0,1,-1,2,-2,3,-3,\cdots \}$$ \BS Cantor was convinced the WOP was true and thought he had proven it. \noi \noi Question: Can $\fatr$ be well-ordered? \noi \noi Under the usual ordering, $(0,1)$ has no least element (nor does $\fatr$ itself!). \noi \noi No well-order for $\fatr$ is known. \noi \noi Does this mean $\fatr$ can't be well-ordered?? \BS Zermelo (1910) thought he too had proven the WOP. However, he had implicitly assumed the validity of the Axiom of Choice. \noi \noi Axiom of Choice (AC): Given an infinite collection of sets, it is possible to choose one element from each set without giving a rule of choice. \noi \noi Russell: {\it If you have an infinite collection of pairs of shoes, then the AC is not needed to identify one from each pair, for certainly one can choose, for example, the left shoe from each pair. However, the AC must be invoked to choose one member from an infinite collection of pairs of socks. Since the socks are identical, no rule can be stated as to how to choose one from each pair} \BS Borel showed that the AC and the WOP were in fact equivalent. \noi \noi Proof that WOP $\Longrightarrow$ AC: \vskip5pt\noindent \noi Take an infinite collection of sets. \noi Assume each is well-ordered, so that each has a least element. \noi Choose the least element. \BS By 1920, set theory was axiomatized to the satisfaction of most mathemticians. However, the AC and CH still were of concern. \noi \noi Attempts were made to prove the AC from the other axioms of set theory. \noi \noi Attempts were made to prove the CH from the other axioms of set theory. \noi \noi \cl {HAH! Nice try} \noi \BS \MYTITLE {Kurt Godel (1940)} \cl{\includegraphics[height=4in,width=3in]{/home/dwick/DATA/Professional/Other/Math/math_photos/Godel.jpg}} \noi \begin{enumerate} \item {} The AC cannot be disproved \noi \item {} The CH cannot be disproved \end{enumerate} \BS \MYTITLE{Paul Cohen (1964)} \cl{\includegraphics[height=4in,width=3in]{/home/dwick/DATA/Professional/Other/Math/math_photos/Cohen.jpg}}} \noi \begin{enumerate} \item {} The AC cannot be proved \noi \item {} The CH cannot be proved \end{enumerate} \noi \BS The AC and CH are undecidable propositions within the current axioms of set theory. \noi Godel had actually shown in 1932 that such undecidable propositions were bound to exist. \noi \noi Dieudonne: {\it It's good to know if something is undecidable. That way we won't waste time trying to decide it} \noi \noi Smullyan: {\it The CH is certainly either true or false, it's just that the current axioms of set theory are not sufficient to allow us to decide} \BS \MYTITLE {Current Status} \noi The Zermelo-Fraenkel axioms of set theory are free of paradox, at least as far as we know! \noi \noi AC is usually added as an axiom to the current axioms of set theory. It's use can, however, lead to some bizarre paradoxes. Consequently, it is not universally accepted by \underbar {all} mathematicians \noi \noi Godel thought CH was not true. It is generally not taken as an axiom of set theory, and is relegated to "undecidable" status \end{document}