{"id":28,"date":"2021-12-15T09:52:57","date_gmt":"2021-12-15T09:52:57","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/zahlenmengen\/"},"modified":"2021-12-15T09:52:57","modified_gmt":"2021-12-15T09:52:57","slug":"zahlenmengen","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/zahlenmengen\/","title":{"raw":"Zahlenmengen","rendered":"Zahlenmengen"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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\n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"zfbfbfde4c686\" class=\"sectionHead\"><span class=\"titlemark\">1.5 <\/span> <a id=\"x1-190005\"><\/a>Zahlenmengen<\/h3> <blockquote class=\"quote\"> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">Die nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlichen Zahlen hat der liebe Gott gemacht,<\/span><br> <span class=\"ecti-1095\">alles andere ist Menschenwerk.<\/span><span class=\"ecti-1095\">\u201c<\/span><br> leicht adaptiert nach Kronecker (1823-1891)<\/p><\/div> <\/blockquote> <p class=\"noindent\">In diesem Abschnitt wollen wir wahrscheinlich schon bekannte Zahlenmengen kurz besprechen. Informell w\u00e4re man wahrscheinlich dazu verleitet, folgende bekannte Mengen zu definieren: <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u2115<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u2124<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u211a<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">:<\/mo> <mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u211d<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211a<\/mi> <mo class=\"MathClass-bin\">\u222a<\/mo><mo class=\"MathClass-open\">{<\/mo><mstyle class=\"text\"><mtext>alle&nbsp;\u201eL\u00fccken\u201c<\/mtext><\/mstyle><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Bereits am Beispiel der Menge der nat\u00fcrlichen Zahlen <math display=\"inline\"><mi>\u2115<\/mi><\/math> l\u00e4sst sich erkennen, dass dies im Sinne der Mathematik keine Definition ist oder sein kann. In der Tat, was bedeuten die Punkte in obigen Formeln genau? Vielleicht deuetet aber obiger \u201eVersuch einer Definition\u201c an, dass jede nat\u00fcrliche Zahl einen Nachfolger besitzen soll. Zum Beispiel ist <math display=\"inline\"><mn>2<\/mn><\/math> der Name des Nachfolgers von <math display=\"inline\"><mn>1<\/mn><\/math> und <math display=\"inline\"><mn>3<\/mn><\/math> der Name des                                                                                                                                                                           Nachfolgers von <span class=\"maperiod\"><math display=\"inline\"><mn>2<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Weiter sollten die nat\u00fcrlichen Zahlen mit einer Addition und einer Multiplikation ausgestattet sein, die die \u00fcblichen Assoziativ- und die Distributivregeln erf\u00fcllen sollten. Formal sind die nat\u00fcrlichen Zahlen durch folgendes Axiomensystem definiert. Die Existenz und Eigenschaften der Addition und Multiplikation kann \u00fcberraschenderweise bereits aus diesem \u00e4ussert minimalen Axiomensystem abgeleitet werden. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z0868b6b291ce\"> <span class=\"ecbx-1095\">Peano Axiome.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Die <\/span><span class=\"ecbi-1095\">nat<\/span><span class=\"ecbi-1095\">\u00fc<\/span><span class=\"ecbi-1095\">rlichen Zahlen <\/span><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">sind (in einem gewissen Sinne eindeutig) durch die folgenden Eigenschaften charakterisiert:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es existiert ein ausgezeichnetes Element <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und eine injektive Abbildung <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03bd<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">auch Nachfolgerfunktion genannt, so dass <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn><mo class=\"MathClass-rel\">\u2209<\/mo><mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt das Induktionsaxiom: Ist <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">enth<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lt und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">die Eigenschaft<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"maendquote\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"endquote\">\u201c<\/span> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt, dann gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\">Die Nachfolgerfunktion <math display=\"inline\"><mi>\u03bd<\/mi><\/math> sollte man sich als die Abbildung <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> vorstellen, nur kennt man die Addition auf <math display=\"inline\"><mi>\u2115<\/mi><\/math> noch nicht. Wir k\u00f6nnen aber die Nachfolgerfunktion verwenden um zum Beispiel <span class=\"maperiod\"><math display=\"inline\"><mn>2<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mn>3<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mn>4<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>3<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mn>5<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>4<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> zu definieren. F\u00fcr eine detailliertere Diskussion dieses Axiomensystems verweisen wir auf <span class=\"cite\">[<a href=\"#Xamann-escher\">AE06<\/a>]<\/span>; wir m\u00f6chten bloss an einem elementaren Beispiel demonstrieren, wie man das Induktionsaxiom verwenden kann. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z3647b02904c8\"> <span class=\"ecbx-1095\">Behauptung <\/span>(Nachfolgerzahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">eine nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl verschieden von <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir betrachten die Menge <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u222a<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> per Definition und f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2115<\/mi><\/math> gilt <math display=\"inline\"><mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> wieder per Definition von <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> Nach dem Induktionsaxiom ist also <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2115<\/mi><\/math> und die Behauptung folgt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Man kann ausgehend von den Peano-Axiomen sowohl Addition und Multiplikation mittels vollst\u00e4ndiger Induktion (oder \u00e4quivalenterweise mittels Rekursion) definieren. Kennt man die nat\u00fcrlichen Zahlen, so lassen sich aus diesen die Zahlenmengen <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>\u2124<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>\u211a<\/mi><\/math> und auch <math display=\"inline\"><mi>\u211d<\/mi><\/math> konstruieren. In dieser Vorlesung werden wir die reellen Zahlen in Kapitel&nbsp;<a href=\"..\/..\/part\/die-reellen-zahlen#x1-430002\">2<\/a> axiomatisch einf\u00fchren und dann die Zahlenmengen <span class=\"maperiod\"><math display=\"inline\"><mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>\u2124<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>\u211a<\/mi><\/math> als Teilmengen von <math display=\"inline\"><mi>\u211d<\/mi><\/math> nochmals ausf\u00fchrlich definieren. Insbesondere werden wir sehen, dass die Menge der reellen Zahlen <math display=\"inline\"><mi>\u211d<\/mi><\/math> nebst den rationalen Zahlen viele weitere Zahlen wie zum Beispiel <span class=\"maperiod\"><math display=\"inline\"><msqrt><mrow> <mn>2<\/mn><\/mrow><\/msqrt><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>\u03c0<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/math><\/span><span class=\"period\">,<\/span> \u2026enth\u00e4lt<button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 Wir werden im Laufe des ersten Semesters all diese Zahlen definieren.<\/span><\/span>. Es wird sich auch herausstellen, dass eine \u201etypische\u201c reelle Zahl nicht rational (also irrational) ist. Historisch gesehen wurden die reellen Zahlen lange verwendet bevor eine rigorose Definition \u00fcberhaupt vorhanden war. Insbesondere wurde ein Grossteil der klassischen Analysis, so wie sie in dieser Vorlesung besprochen wird, vor der ersten l\u00fcckenfreien Definition der reellen Zahlen durch Cantor im zweiten Teil des 19ten Jahrhunderts entwickelt.                                                                                                                                                                           <\/p><p class=\"indent\">Wie bereits im letzten Abschnitt wollen wir auch im Rest des Kapitels annehmen, dass wir die \u00fcblichen Zahlenmengen mit allen \u00fcblichen Eigenschaften bereits kennen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z2193ca88832d\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Eigentlich lautet das Zitat (siehe Seite 15 in <span class=\"cite\">[<a href=\"#Xkroneckerzitat\">Web92<\/a>]<\/span>) von Kronecker zu Beginn dieses Abschnitts <\/p><blockquote class=\"quote\"> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">Die <\/span><span class=\"underline\"><span class=\"ecti-1095\">ganzen<\/span> <\/span> <span class=\"ecti-1095\">Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.<\/span><span class=\"ecti-1095\">\u201c<\/span><\/p><\/div> <\/blockquote> <p class=\"noindent\">Es kann jedoch sein, dass Kronecker eigentlich die nat\u00fcrlichen Zahlen gemeint hat. Auf jeden Fall wollen wir hier den M\u00f6nch und Historiker William of Malmesbury (ca. 1095-1143), der die ganze Zahl <math display=\"inline\"><mn>0<\/mn><\/math> als \u201e dangerous Saracen magic\u201c bezeichnet hat, als gr\u00f6ssere Autorit\u00e4t in dieser Frage ansehen. Scherz beiseite, die Menschheit (und inbesondere Europa) hat in der Tat sehr lange gebraucht, um mit der Null und den negativen Zahlen zurecht zu kommen. F\u00fcr einen geschichtlichen Exkurs zur \u201eZahl <span class=\"maendquote\"><math display=\"inline\"><mn>0<\/mn><\/math><\/span><span class=\"endquote\">\u201c<\/span> verweisen wir auf diesen <a href=\"http:\/\/www.bbc.co.uk\/programmes\/p004y254\" target=\"_blank\" rel=\"noopener\">Podcast<\/a> der BBC, und zum Thema \u201eNegative Zahlen\u201c auf die ersten <math display=\"inline\"><mn>1<\/mn><mn>0<\/mn><\/math>-<math display=\"inline\"><mn>2<\/mn><mn>0<\/mn><\/math> Minuten eines weiteren <a href=\"http:\/\/www.bbc.co.uk\/programmes\/p003hyd9\" target=\"_blank\" rel=\"noopener\">Podcasts<\/a> der BBC. <\/p> <\/div> <p class=\"indent\"> <a id=\"x1-19003r19\"><\/a> <\/p> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"zfbfbfde4c686\" class=\"sectionHead\"><span class=\"titlemark\">1.5 <\/span> <a id=\"x1-190005\"><\/a>Zahlenmengen<\/h3> <blockquote class=\"quote\"> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><p class=\"noindent\"><span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">Die nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlichen Zahlen hat der liebe Gott gemacht,<\/span><br \/> <span class=\"ecti-1095\">alles andere ist Menschenwerk.<\/span><span class=\"ecti-1095\">\u201c<\/span><br \/> leicht adaptiert nach Kronecker (1823-1891)<\/p><\/div> <\/blockquote> <p class=\"noindent\">In diesem Abschnitt wollen wir wahrscheinlich schon bekannte Zahlenmengen kurz besprechen. Informell w\u00e4re man wahrscheinlich dazu verleitet, folgende bekannte Mengen zu definieren: <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u2115<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u2124<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u211a<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">:<\/mo> <mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u211d<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u211a<\/mi> <mo class=\"MathClass-bin\">\u222a<\/mo><mo class=\"MathClass-open\">{<\/mo><mstyle class=\"text\"><mtext>alle&nbsp;\u201eL\u00fccken\u201c<\/mtext><\/mstyle><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Bereits am Beispiel der Menge der nat\u00fcrlichen Zahlen <math display=\"inline\"><mi>\u2115<\/mi><\/math> l\u00e4sst sich erkennen, dass dies im Sinne der Mathematik keine Definition ist oder sein kann. In der Tat, was bedeuten die Punkte in obigen Formeln genau? Vielleicht deuetet aber obiger \u201eVersuch einer Definition\u201c an, dass jede nat\u00fcrliche Zahl einen Nachfolger besitzen soll. Zum Beispiel ist <math display=\"inline\"><mn>2<\/mn><\/math> der Name des Nachfolgers von <math display=\"inline\"><mn>1<\/mn><\/math> und <math display=\"inline\"><mn>3<\/mn><\/math> der Name des                                                                                                                                                                           Nachfolgers von <span class=\"maperiod\"><math display=\"inline\"><mn>2<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Weiter sollten die nat\u00fcrlichen Zahlen mit einer Addition und einer Multiplikation ausgestattet sein, die die \u00fcblichen Assoziativ- und die Distributivregeln erf\u00fcllen sollten. Formal sind die nat\u00fcrlichen Zahlen durch folgendes Axiomensystem definiert. Die Existenz und Eigenschaften der Addition und Multiplikation kann \u00fcberraschenderweise bereits aus diesem \u00e4ussert minimalen Axiomensystem abgeleitet werden. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z0868b6b291ce\"> <span class=\"ecbx-1095\">Peano Axiome.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Die <\/span><span class=\"ecbi-1095\">nat<\/span><span class=\"ecbi-1095\">\u00fc<\/span><span class=\"ecbi-1095\">rlichen Zahlen <\/span><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">sind (in einem gewissen Sinne eindeutig) durch die folgenden Eigenschaften charakterisiert:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es existiert ein ausgezeichnetes Element <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und eine injektive Abbildung <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03bd<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">auch Nachfolgerfunktion genannt, so dass <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn><mo class=\"MathClass-rel\">\u2209<\/mo><mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt das Induktionsaxiom: Ist <\/span><math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">enth<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lt und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">die Eigenschaft<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"maendquote\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"endquote\">\u201c<\/span> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt, dann gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\">Die Nachfolgerfunktion <math display=\"inline\"><mi>\u03bd<\/mi><\/math> sollte man sich als die Abbildung <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> vorstellen, nur kennt man die Addition auf <math display=\"inline\"><mi>\u2115<\/mi><\/math> noch nicht. Wir k\u00f6nnen aber die Nachfolgerfunktion verwenden um zum Beispiel <span class=\"maperiod\"><math display=\"inline\"><mn>2<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mn>3<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mn>4<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>3<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mn>5<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>4<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> zu definieren. F\u00fcr eine detailliertere Diskussion dieses Axiomensystems verweisen wir auf <span class=\"cite\">[<a href=\"#Xamann-escher\">AE06<\/a>]<\/span>; wir m\u00f6chten bloss an einem elementaren Beispiel demonstrieren, wie man das Induktionsaxiom verwenden kann. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z3647b02904c8\"> <span class=\"ecbx-1095\">Behauptung <\/span>(Nachfolgerzahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">eine nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl verschieden von <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir betrachten die Menge <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2115<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u222a<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> per Definition und f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2115<\/mi><\/math> gilt <math display=\"inline\"><mi>\u03bd<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math> wieder per Definition von <span class=\"maperiod\"><math display=\"inline\"><mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> Nach dem Induktionsaxiom ist also <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2115<\/mi><\/math> und die Behauptung folgt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Man kann ausgehend von den Peano-Axiomen sowohl Addition und Multiplikation mittels vollst\u00e4ndiger Induktion (oder \u00e4quivalenterweise mittels Rekursion) definieren. Kennt man die nat\u00fcrlichen Zahlen, so lassen sich aus diesen die Zahlenmengen <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>\u2124<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>\u211a<\/mi><\/math> und auch <math display=\"inline\"><mi>\u211d<\/mi><\/math> konstruieren. In dieser Vorlesung werden wir die reellen Zahlen in Kapitel&nbsp;<a href=\"..\/..\/part\/die-reellen-zahlen#x1-430002\">2<\/a> axiomatisch einf\u00fchren und dann die Zahlenmengen <span class=\"maperiod\"><math display=\"inline\"><mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>\u2124<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>\u211a<\/mi><\/math> als Teilmengen von <math display=\"inline\"><mi>\u211d<\/mi><\/math> nochmals ausf\u00fchrlich definieren. Insbesondere werden wir sehen, dass die Menge der reellen Zahlen <math display=\"inline\"><mi>\u211d<\/mi><\/math> nebst den rationalen Zahlen viele weitere Zahlen wie zum Beispiel <span class=\"maperiod\"><math display=\"inline\"><msqrt><mrow> <mn>2<\/mn><\/mrow><\/msqrt><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>\u03c0<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/math><\/span><span class=\"period\">,<\/span> \u2026enth\u00e4lt<button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 Wir werden im Laufe des ersten Semesters all diese Zahlen definieren.<\/span><\/span>. Es wird sich auch herausstellen, dass eine \u201etypische\u201c reelle Zahl nicht rational (also irrational) ist. Historisch gesehen wurden die reellen Zahlen lange verwendet bevor eine rigorose Definition \u00fcberhaupt vorhanden war. Insbesondere wurde ein Grossteil der klassischen Analysis, so wie sie in dieser Vorlesung besprochen wird, vor der ersten l\u00fcckenfreien Definition der reellen Zahlen durch Cantor im zweiten Teil des 19ten Jahrhunderts entwickelt.                                                                                                                                                                           <\/p><p class=\"indent\">Wie bereits im letzten Abschnitt wollen wir auch im Rest des Kapitels annehmen, dass wir die \u00fcblichen Zahlenmengen mit allen \u00fcblichen Eigenschaften bereits kennen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2193ca88832d\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Eigentlich lautet das Zitat (siehe Seite 15 in <span class=\"cite\">[<a href=\"#Xkroneckerzitat\">Web92<\/a>]<\/span>) von Kronecker zu Beginn dieses Abschnitts <\/p><blockquote class=\"quote\"> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><p class=\"noindent\"><span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">Die <\/span><span class=\"underline\"><span class=\"ecti-1095\">ganzen<\/span> <\/span> <span class=\"ecti-1095\">Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.<\/span><span class=\"ecti-1095\">\u201c<\/span><\/p><\/div> <\/blockquote> <p class=\"noindent\">Es kann jedoch sein, dass Kronecker eigentlich die nat\u00fcrlichen Zahlen gemeint hat. Auf jeden Fall wollen wir hier den M\u00f6nch und Historiker William of Malmesbury (ca. 1095-1143), der die ganze Zahl <math display=\"inline\"><mn>0<\/mn><\/math> als \u201e dangerous Saracen magic\u201c bezeichnet hat, als gr\u00f6ssere Autorit\u00e4t in dieser Frage ansehen. Scherz beiseite, die Menschheit (und inbesondere Europa) hat in der Tat sehr lange gebraucht, um mit der Null und den negativen Zahlen zurecht zu kommen. F\u00fcr einen geschichtlichen Exkurs zur \u201eZahl <span class=\"maendquote\"><math display=\"inline\"><mn>0<\/mn><\/math><\/span><span class=\"endquote\">\u201c<\/span> verweisen wir auf diesen <a href=\"http:\/\/www.bbc.co.uk\/programmes\/p004y254\" target=\"_blank\" rel=\"noopener\">Podcast<\/a> der BBC, und zum Thema \u201eNegative Zahlen\u201c auf die ersten <math display=\"inline\"><mn>1<\/mn><mn>0<\/mn><\/math>&#8211;<math display=\"inline\"><mn>2<\/mn><mn>0<\/mn><\/math> Minuten eines weiteren <a href=\"http:\/\/www.bbc.co.uk\/programmes\/p003hyd9\" target=\"_blank\" rel=\"noopener\">Podcasts<\/a> der BBC. <\/p> <\/div> <p class=\"indent\"> <a id=\"x1-19003r19\"><\/a> <\/p> \n","protected":false},"author":1089,"menu_order":5,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-28","chapter","type-chapter","status-publish","hentry"],"part":23,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/28","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/users\/1089"}],"version-history":[{"count":0,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/28\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/23"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/28\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=28"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=28"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=28"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=28"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}