Pourquoi l’infini a plusieurs tailles \u2014 MathsVivantes<\/title>\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Playfair+Display:ital,wght@0,400;0,700;1,400&family=Source+Serif+4:ital,wght@0,300;0,400;0,600;1,300;1,400&display=swap\" rel=\"stylesheet\">\n<style>\n :root {\n --noir: #0f0e0b;\n --creme: #f5f0e8;\n --cosmos: #0a0520;\n --violet: #5a1a9a;\n --violet-clair: #9a4adf;\n --violet-pale: #f5eeff;\n --or: #c8a010;\n --or-clair: #f0d060;\n --rouge: #b03a2e;\n --gris: #6b6560;\n --gris-clair: #e8e2d8;\n }\n\n * { margin: 0; padding: 0; box-sizing: border-box; }\n\n body {\n background-color: var(--creme);\n color: var(--noir);\n font-family: 'Source Serif 4', Georgia, serif;\n font-weight: 300;\n line-height: 1.8;\n }\n\n .hero {\n background: #04010f;\n color: var(--creme);\n padding: 80px 40px 70px;\n position: relative;\n overflow: hidden;\n }\n .hero::before {\n content: \"\u221e \u2135\u2080 \u2135\u2081 \u221e\";\n position: absolute;\n top: 20px; right: 30px;\n font-size: 5rem;\n opacity: 0.05;\n font-family: 'Playfair Display', serif;\n letter-spacing: 0.3em;\n }\n .hero-inner { max-width: 760px; margin: 0 auto; }\n\n .article-rubrique {\n display: inline-block;\n background: var(--violet);\n color: white;\n font-size: 0.7rem;\n letter-spacing: 0.18em;\n text-transform: uppercase;\n padding: 4px 12px;\n margin-bottom: 24px;\n font-weight: 600;\n }\n\n .hero h1 {\n font-family: 'Playfair Display', serif;\n font-size: clamp(2rem, 5vw, 3.2rem);\n line-height: 1.2;\n margin-bottom: 24px;\n font-weight: 700;\n }\n .hero h1 em { color: var(--or-clair); font-style: italic; }\n\n .hero-intro {\n font-size: 1.15rem;\n opacity: 0.75;\n max-width: 600px;\n font-style: italic;\n border-left: 3px solid var(--violet-clair);\n padding-left: 20px;\n margin-bottom: 32px;\n }\n .meta { font-size: 0.8rem; opacity: 0.45; letter-spacing: 0.08em; text-transform: uppercase; }\n\n .article-body { max-width: 760px; margin: 0 auto; padding: 60px 40px 80px; }\n .article-body p { font-size: 1.05rem; margin-bottom: 1.6em; color: #2a2520; }\n\n h2 {\n font-family: 'Playfair Display', serif;\n font-size: 1.6rem;\n margin: 2.5em 0 0.8em;\n color: var(--noir);\n font-weight: 700;\n }\n h2::before { content: \"\u00a7 \"; color: var(--violet-clair); }\n\n h3 {\n font-family: 'Playfair Display', serif;\n font-size: 1.15rem;\n font-weight: 700;\n margin: 1.8em 0 0.6em;\n color: var(--violet);\n }\n\n .encadre {\n border: 1.5px solid var(--violet-clair);\n background: white;\n padding: 24px 28px;\n margin: 2em 0;\n position: relative;\n }\n .encadre-titre {\n position: absolute;\n top: -12px; left: 20px;\n background: white;\n padding: 0 8px;\n font-size: 0.72rem;\n letter-spacing: 0.15em;\n text-transform: uppercase;\n color: var(--violet-clair);\n font-weight: 600;\n }\n .encadre p { margin-bottom: 0.8em; font-size: 0.98rem; }\n .encadre p:last-child { margin-bottom: 0; }\n\n .encadre-histoire {\n background: #04010f;\n color: var(--creme);\n border: none;\n padding: 32px 36px;\n margin: 2.5em 0;\n }\n .encadre-histoire .encadre-titre { background: #04010f; color: var(--or-clair); }\n .encadre-histoire p { color: rgba(245,240,232,0.85); }\n\n .definition {\n background: var(--violet-pale);\n border-left: 4px solid var(--violet-clair);\n padding: 20px 24px;\n margin: 1.8em 0;\n font-style: italic;\n }\n .definition strong { font-style: normal; color: var(--violet); font-family: 'Playfair Display', serif; }\n\n .choc {\n background: #04010f;\n color: var(--creme);\n padding: 28px 32px;\n margin: 2em 0;\n text-align: center;\n }\n .choc-titre {\n font-size: 0.72rem;\n letter-spacing: 0.18em;\n text-transform: uppercase;\n color: var(--violet-clair);\n margin-bottom: 12px;\n font-weight: 600;\n }\n .choc-contenu {\n font-family: 'Playfair Display', serif;\n font-size: 1.3rem;\n color: var(--or-clair);\n line-height: 1.6;\n }\n\n .schema-container {\n background: white;\n border: 1.5px solid var(--violet-clair);\n padding: 24px;\n margin: 2em 0;\n text-align: center;\n }\n .schema-titre {\n font-size: 0.72rem;\n letter-spacing: 0.15em;\n text-transform: uppercase;\n color: var(--violet-clair);\n font-weight: 600;\n margin-bottom: 16px;\n }\n\n .formule {\n text-align: center;\n font-size: 1.25rem;\n font-family: 'Playfair Display', serif;\n margin: 1.5em 0;\n color: var(--noir);\n padding: 16px;\n background: white;\n border-top: 2px solid var(--violet-clair);\n border-bottom: 2px solid var(--violet-clair);\n }\n\n .etape {\n background: white;\n border-left: 3px solid var(--violet-clair);\n padding: 14px 20px;\n margin: 0.6em 0;\n font-size: 0.97rem;\n }\n .etape-numero {\n font-family: 'Playfair Display', serif;\n font-weight: 700;\n color: var(--violet-clair);\n margin-right: 8px;\n }\n\n \/* HI\u00c9RARCHIE DES INFINIS *\/\n .infinis-scale {\n margin: 2em 0;\n display: flex;\n flex-direction: column;\n gap: 8px;\n }\n .infini-row {\n display: flex;\n align-items: center;\n gap: 16px;\n }\n .infini-label {\n min-width: 80px;\n font-family: 'Playfair Display', serif;\n font-size: 1.1rem;\n font-weight: 700;\n text-align: right;\n }\n .infini-bar {\n height: 36px;\n display: flex;\n align-items: center;\n padding: 0 16px;\n font-size: 0.85rem;\n color: white;\n font-weight: 600;\n border-radius: 2px;\n }\n .infini-desc {\n font-size: 0.85rem;\n color: var(--gris);\n font-style: italic;\n }\n\n .code-block {\n background: #04010f;\n color: #d0b0f0;\n padding: 24px 28px;\n margin: 2em 0;\n font-family: 'Courier New', monospace;\n font-size: 0.88rem;\n line-height: 1.7;\n overflow-x: auto;\n position: relative;\n }\n .code-label { position: absolute; top: 8px; right: 12px; font-size: 0.65rem; letter-spacing: 0.15em; text-transform: uppercase; color: var(--violet-clair); font-family: 'Source Serif 4', serif; }\n .code-comment { color: #5a3a8a; }\n .code-keyword { color: #a080d0; }\n .code-string { color: var(--or-clair); }\n .code-number { color: #c0a0f0; }\n .code-function { color: #e0c0ff; }\n\n .exercice {\n background: #04010f;\n color: var(--creme);\n padding: 36px 40px;\n margin: 3em 0;\n }\n .exercice h3 { color: var(--or-clair); font-style: normal; margin-top: 0; margin-bottom: 1em; font-size: 1.1rem; font-weight: 600; }\n .exercice p { color: rgba(245,240,232,0.82); font-size: 0.97rem; margin-bottom: 0.8em; }\n .exercice .correction { border-top: 1px solid rgba(90,26,154,0.4); margin-top: 1.5em; padding-top: 1.2em; }\n .exercice .correction-titre { font-size: 0.7rem; letter-spacing: 0.18em; text-transform: uppercase; color: var(--or-clair); margin-bottom: 0.8em; font-weight: 600; }\n\n blockquote {\n padding: 0 0 0 32px;\n margin: 2em 0;\n position: relative;\n font-style: italic;\n font-size: 1.12rem;\n color: #3a3530;\n }\n blockquote::before {\n content: \"\\201C\";\n position: absolute;\n left: 0; top: -10px;\n font-size: 3.5rem;\n color: var(--violet-clair);\n font-family: 'Playfair Display', serif;\n line-height: 1;\n }\n blockquote cite { display: block; font-size: 0.82rem; font-style: normal; letter-spacing: 0.1em; text-transform: uppercase; color: var(--gris); margin-top: 8px; }\n\n .conclusion { border-top: 2px solid var(--noir); margin-top: 3em; padding-top: 2em; }\n\n @media (max-width: 600px) {\n .hero { padding: 50px 20px 40px; }\n .article-body { padding: 40px 20px 60px; }\n .exercice { padding: 24px 20px; }\n .code-block { padding: 16px; font-size: 0.78rem; }\n .infini-row { flex-wrap: wrap; }\n }\n<\/style>\n<\/head>\n<body>\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">Philosophie des maths \u00b7 Article #15<\/div>\n <h1>Pourquoi l’infini<br>a <em>plusieurs tailles<\/em> \u2014<br>et certains sont plus grands<\/h1>\n <p class=\"hero-intro\">Combien y a-t-il de nombres entiers ? Infini. Combien y a-t-il de nombres r\u00e9els ? Aussi infini. Mais ces deux infinis ne sont pas \u00e9gaux. L’un est strictement plus grand que l’autre. Et c’est d\u00e9montrable. Bienvenue dans l’une des id\u00e9es les plus vertigineuses de toute l’histoire des math\u00e9matiques.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 13 min<\/div>\n <\/div>\n<\/div>\n\n<article class=\"article-body\">\n\n <p>\n Imagine que tu comptes les grains de sable de la plage de Fidjross\u00e8 \u00e0 Cotonou. Tu n’y arriveras jamais \u2014 il y en a trop. Maintenant imagine que tu comptes les \u00e9toiles de l’univers. Encore plus impossible. Ces deux quantit\u00e9s sont toutes les deux \u00ab\u00a0infinies\u00a0\u00bb dans le sens pratique \u2014 mais sont-elles <em>\u00e9gales<\/em> ?\n <\/p>\n\n <p>\n Un math\u00e9maticien allemand nomm\u00e9 Georg Cantor a pos\u00e9 cette question au XIXe si\u00e8cle et a obtenu une r\u00e9ponse qui a failli le rendre fou \u2014 et qui a choqu\u00e9 la communaut\u00e9 math\u00e9matique mondiale. Sa r\u00e9ponse : <strong>non, tous les infinis ne sont pas \u00e9gaux. Certains infinis sont strictement plus grands que d’autres.<\/strong>\n <\/p>\n\n <p>\n Et il l’a <em>prouv\u00e9<\/em>.\n <\/p>\n\n <h2>Commen\u00e7ons par le commencement : compter sans nombres<\/h2>\n\n <p>\n Avant d’attaquer l’infini, il faut r\u00e9apprendre ce que signifie \u00ab\u00a0m\u00eame taille\u00a0\u00bb. La d\u00e9finition intuitive \u2014 compter les \u00e9l\u00e9ments et comparer \u2014 ne fonctionne pas pour des ensembles infinis. Il faut une d\u00e9finition plus profonde.\n <\/p>\n\n <div class=\"definition\">\n <strong>Bijection (ou correspondance biunivoque) :<\/strong> Une bijection entre deux ensembles A et B est une r\u00e8gle qui associe \u00e0 chaque \u00e9l\u00e9ment de A exactement un \u00e9l\u00e9ment de B, et vice versa \u2014 sans rien oublier, sans rien r\u00e9p\u00e9ter. Si une telle bijection existe, les deux ensembles ont la <em>m\u00eame taille<\/em> \u2014 le m\u00eame <strong>cardinal<\/strong>.\n <\/div>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83c\udf0d Analogie du march\u00e9 de Dantokpa<\/span>\n <p>Imagine deux files au march\u00e9 : une file de vendeuses et une file d’acheteurs. Pour savoir si les deux files ont le m\u00eame nombre de personnes, tu n’as pas besoin de les compter. Il suffit de les <strong>jumeler<\/strong> \u2014 chaque vendeuse face \u00e0 un acheteur. S’il n’y a ni vendeuse ni acheteur sans partenaire \u00e0 la fin, les deux files ont le m\u00eame effectif.<\/p>\n <p>C’est exactement la d\u00e9finition d’une bijection. Et c’est le seul outil rigoureux pour comparer des ensembles infinis.<\/p>\n <\/div>\n\n <h2>Premier choc : les nombres pairs sont \u00ab\u00a0aussi nombreux\u00a0\u00bb que les entiers<\/h2>\n\n <p>\n Voici quelque chose d’imm\u00e9diatement contre-intuitif. Les nombres pairs (2, 4, 6, 8…) semblent deux fois moins nombreux que les entiers (1, 2, 3, 4…). Pourtant, en math\u00e9matiques rigoureuses, ils ont le <em>m\u00eame cardinal<\/em>.\n <\/p>\n\n <p>Preuve : construisons une bijection entre \u2115 (entiers naturels) et les nombres pairs :<\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udd17 Bijection entre entiers et nombres pairs<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 160\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"160\" fill=\"white\"\/>\n\n \n <text x=\"30\" y=\"40\" font-size=\"12\" fill=\"#5a1a9a\" font-weight=\"bold\">\u2115 :<\/text>\n <rect x=\"60\" y=\"22\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#f5eeff\" stroke=\"#9a4adf\" stroke-width=\"1.5\"\/>\n <text x=\"85\" y=\"41\" font-size=\"13\" fill=\"#5a1a9a\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">1<\/text>\n <rect x=\"120\" y=\"22\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#f5eeff\" stroke=\"#9a4adf\" stroke-width=\"1.5\"\/>\n <text x=\"145\" y=\"41\" font-size=\"13\" fill=\"#5a1a9a\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">2<\/text>\n <rect x=\"180\" y=\"22\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#f5eeff\" stroke=\"#9a4adf\" stroke-width=\"1.5\"\/>\n <text x=\"205\" y=\"41\" font-size=\"13\" fill=\"#5a1a9a\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">3<\/text>\n <rect x=\"240\" y=\"22\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#f5eeff\" stroke=\"#9a4adf\" stroke-width=\"1.5\"\/>\n <text x=\"265\" y=\"41\" font-size=\"13\" fill=\"#5a1a9a\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">4<\/text>\n <rect x=\"300\" y=\"22\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#f5eeff\" stroke=\"#9a4adf\" stroke-width=\"1.5\"\/>\n <text x=\"325\" y=\"41\" font-size=\"13\" fill=\"#5a1a9a\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">5<\/text>\n <text x=\"370\" y=\"41\" font-size=\"16\" fill=\"#9a4adf\">\u2192 \u221e<\/text>\n\n \n <line x1=\"85\" y1=\"50\" x2=\"85\" y2=\"88\" stroke=\"#9a4adf\" stroke-width=\"1.5\" marker-end=\"url(#arrV)\"\/>\n <line x1=\"145\" y1=\"50\" x2=\"145\" y2=\"88\" stroke=\"#9a4adf\" stroke-width=\"1.5\" marker-end=\"url(#arrV)\"\/>\n <line x1=\"205\" y1=\"50\" x2=\"205\" y2=\"88\" stroke=\"#9a4adf\" stroke-width=\"1.5\" marker-end=\"url(#arrV)\"\/>\n <line x1=\"265\" y1=\"50\" x2=\"265\" y2=\"88\" stroke=\"#9a4adf\" stroke-width=\"1.5\" marker-end=\"url(#arrV)\"\/>\n <line x1=\"325\" y1=\"50\" x2=\"325\" y2=\"88\" stroke=\"#9a4adf\" stroke-width=\"1.5\" marker-end=\"url(#arrV)\"\/>\n <defs>\n <marker id=\"arrV\" markerWidth=\"7\" markerHeight=\"7\" refX=\"5\" refY=\"3\" orient=\"auto\">\n <path d=\"M0,0 L0,6 L7,3 z\" fill=\"#9a4adf\"\/>\n <\/marker>\n <\/defs>\n\n \n <text x=\"440\" y=\"72\" font-size=\"11\" fill=\"#9a4adf\" font-style=\"italic\">n \u2192 2n<\/text>\n\n \n <text x=\"30\" y=\"118\" font-size=\"12\" fill=\"#5a1a9a\" font-weight=\"bold\">Pairs :<\/text>\n <rect x=\"60\" y=\"100\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#5a1a9a\"\/>\n <text x=\"85\" y=\"119\" font-size=\"13\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">2<\/text>\n <rect x=\"120\" y=\"100\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#5a1a9a\"\/>\n <text x=\"145\" y=\"119\" font-size=\"13\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">4<\/text>\n <rect x=\"180\" y=\"100\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#5a1a9a\"\/>\n <text x=\"205\" y=\"119\" font-size=\"13\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">6<\/text>\n <rect x=\"240\" y=\"100\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#5a1a9a\"\/>\n <text x=\"265\" y=\"119\" font-size=\"13\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">8<\/text>\n <rect x=\"300\" y=\"100\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#5a1a9a\"\/>\n <text x=\"325\" y=\"119\" font-size=\"13\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">10<\/text>\n <text x=\"370\" y=\"119\" font-size=\"16\" fill=\"#5a1a9a\">\u2192 \u221e<\/text>\n\n \n <text x=\"260\" y=\"150\" font-size=\"11\" fill=\"#5a1a9a\" text-anchor=\"middle\">Bijection parfaite : m\u00eame cardinal !<\/text>\n <\/svg>\n <\/div>\n\n <p>\n La r\u00e8gle n \u2192 2n est une bijection parfaite : chaque entier correspond exactement \u00e0 un pair, et chaque pair vient d’exactement un entier. Donc les entiers et les pairs ont le <strong>m\u00eame cardinal<\/strong> \u2014 m\u00eame si les pairs semblent \u00ab\u00a0moins nombreux\u00a0\u00bb.\n <\/p>\n\n <p>\n C’est le premier signe que l’infini ne se comporte pas comme les nombres finis. Avec l’infini, une partie peut \u00eatre aussi grande que le tout. C’est contre-intuitif \u2014 et c’est rigoureusement vrai.\n <\/p>\n\n <h2>Les infinis d\u00e9nombrables \u2014 le premier niveau<\/h2>\n\n <div class=\"definition\">\n <strong>Ensemble d\u00e9nombrable :<\/strong> Un ensemble dont les \u00e9l\u00e9ments peuvent \u00eatre mis en bijection avec les entiers naturels \u2115. Autrement dit, on peut les \u00ab\u00a0lister\u00a0\u00bb \u2014 m\u00eame si la liste est infinie. Le cardinal de \u2115 se note \u2135\u2080 (aleph-z\u00e9ro).\n <\/div>\n\n <p>\n Voici des ensembles qui sont tous d\u00e9nombrables \u2014 ils ont tous le m\u00eame cardinal \u2135\u2080 :\n <\/p>\n\n <div class=\"infinis-scale\">\n <div class=\"infini-row\">\n <div class=\"infini-label\" style=\"color:#5a1a9a;\">\u2115<\/div>\n <div class=\"infini-bar\" style=\"width:80%; background:#5a1a9a;\">0, 1, 2, 3, 4, 5, 6…<\/div>\n <div class=\"infini-desc\">Entiers naturels \u2014 la d\u00e9finition<\/div>\n <\/div>\n <div class=\"infini-row\">\n <div class=\"infini-label\" style=\"color:#7a2aba;\">\u2124<\/div>\n <div class=\"infini-bar\" style=\"width:80%; background:#7a2aba;\">…-2, -1, 0, 1, 2…<\/div>\n <div class=\"infini-desc\">Entiers relatifs \u2014 aussi d\u00e9nombrables !<\/div>\n <\/div>\n <div class=\"infini-row\">\n <div class=\"infini-label\" style=\"color:#9a4adf;\">\u211a<\/div>\n <div class=\"infini-bar\" style=\"width:80%; background:#9a4adf;\">1\/2, 3\/4, 22\/7… toutes les fractions<\/div>\n <div class=\"infini-desc\">Rationnels \u2014 aussi d\u00e9nombrables ! (Cantor l’a prouv\u00e9)<\/div>\n <\/div>\n <\/div>\n\n <p>\n Oui \u2014 il y a autant de fractions que d’entiers naturels. Cantor a d\u00e9montr\u00e9 cela avec une astuce g\u00e9niale : parcourir les fractions en diagonale dans un tableau infini. C’est ce qu’on appelle le <strong>d\u00e9nombrement diagonal<\/strong>.\n <\/p>\n\n <h2>Deuxi\u00e8me choc : les r\u00e9els ne sont PAS d\u00e9nombrables<\/h2>\n\n <p>\n Maintenant, qu’en est-il des nombres r\u00e9els \u211d \u2014 tous les nombres de la droite num\u00e9rique, rationnels et irrationnels confondus ? Sont-ils d\u00e9nombrables ?\n <\/p>\n\n <p>\n Cantor a r\u00e9pondu avec une d\u00e9monstration d’une beaut\u00e9 et d’une simplicit\u00e9 d\u00e9vastatrices. C’est la <strong>diagonale de Cantor<\/strong> \u2014 consid\u00e9r\u00e9e comme l’une des plus belles preuves de toute l’histoire des math\u00e9matiques.\n <\/p>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83e\udde0 Georg Cantor \u2014 l’homme qui a os\u00e9 compter l’infini<\/span>\n <p>\n Georg Cantor (1845\u20131918) \u00e9tait un math\u00e9maticien russo-allemand. Son travail sur les ensembles infinis a \u00e9t\u00e9 r\u00e9volutionnaire \u2014 et sa vie, tragique. Ses id\u00e9es ont \u00e9t\u00e9 violemment rejet\u00e9es par les math\u00e9maticiens de son \u00e9poque, notamment son ancien professeur Leopold Kronecker qui disait de lui qu’il \u00e9tait un \u00ab\u00a0corrupteur de la jeunesse\u00a0\u00bb.\n <\/p>\n <p>\n Cantor a souffert de d\u00e9pression s\u00e9v\u00e8re toute sa vie \u2014 en partie \u00e0 cause du rejet de ses travaux. Il est mort dans un sanatorium en 1918. Moins de 20 ans plus tard, ses id\u00e9es sur l’infini sont devenues les fondements de toute la math\u00e9matique moderne. David Hilbert, le plus grand math\u00e9maticien de l’\u00e9poque, a dit : <em>\u00ab\u00a0Personne ne nous chassera du paradis que Cantor a cr\u00e9\u00e9.\u00a0\u00bb<\/em>\n <\/p>\n <p>\n Ce n’est pas la premi\u00e8re fois dans l’histoire qu’un g\u00e9nie incompris par ses contemporains a d\u00fb attendre la post\u00e9rit\u00e9 pour \u00eatre reconnu. Galil\u00e9e, Boltzmann, Abel \u2014 la liste est longue.\n <\/p>\n <\/div>\n\n <h2>L’argument diagonal de Cantor \u2014 la preuve compl\u00e8te<\/h2>\n\n <p>\n Nous allons prouver que les nombres r\u00e9els entre 0 et 1 ne sont <em>pas<\/em> d\u00e9nombrables. La m\u00e9thode : on suppose qu’ils le sont, et on montre une contradiction.\n <\/p>\n\n <p>\n <strong>Hypoth\u00e8se :<\/strong> Supposons qu’on peut lister tous les r\u00e9els entre 0 et 1 \u2014 une liste infinie mais compl\u00e8te. Appelons-les r\u2081, r\u2082, r\u2083, r\u2084…\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcd0 L’argument diagonal de Cantor<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 300\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"300\" fill=\"white\"\/>\n\n \n <text x=\"20\" y=\"35\" font-size=\"12\" fill=\"#5a1a9a\" font-weight=\"bold\" font-family=\"Playfair Display, serif\">La liste suppos\u00e9e compl\u00e8te :<\/text>\n\n \n <text x=\"20\" y=\"65\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">r\u2081 = 0.<\/text>\n <rect x=\"85\" y=\"51\" width=\"18\" height=\"20\" rx=\"2\" fill=\"#9a4adf\" opacity=\"0.3\"\/>\n <text x=\"85\" y=\"65\" font-size=\"12\" fill=\"#5a1a9a\" font-family=\"Courier New, monospace\" font-weight=\"bold\">4<\/text>\n <text x=\"103\" y=\"65\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">1 5 9 2 6…<\/text>\n\n \n <text x=\"20\" y=\"95\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">r\u2082 = 0.1<\/text>\n <rect x=\"121\" y=\"81\" width=\"18\" height=\"20\" rx=\"2\" fill=\"#9a4adf\" opacity=\"0.3\"\/>\n <text x=\"121\" y=\"95\" font-size=\"12\" fill=\"#5a1a9a\" font-family=\"Courier New, monospace\" font-weight=\"bold\">6<\/text>\n <text x=\"139\" y=\"95\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">2 1 8 2…<\/text>\n\n \n <text x=\"20\" y=\"125\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">r\u2083 = 0.26<\/text>\n <rect x=\"157\" y=\"111\" width=\"18\" height=\"20\" rx=\"2\" fill=\"#9a4adf\" opacity=\"0.3\"\/>\n <text x=\"157\" y=\"125\" font-size=\"12\" fill=\"#5a1a9a\" font-family=\"Courier New, monospace\" font-weight=\"bold\">5<\/text>\n <text x=\"175\" y=\"125\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">3 5 8…<\/text>\n\n \n <text x=\"20\" y=\"155\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">r\u2084 = 0.589<\/text>\n <rect x=\"193\" y=\"141\" width=\"18\" height=\"20\" rx=\"2\" fill=\"#9a4adf\" opacity=\"0.3\"\/>\n <text x=\"193\" y=\"155\" font-size=\"12\" fill=\"#5a1a9a\" font-family=\"Courier New, monospace\" font-weight=\"bold\">7<\/text>\n <text x=\"211\" y=\"155\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">3 2…<\/text>\n\n \n <text x=\"20\" y=\"185\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">r\u2085 = 0.7314<\/text>\n <rect x=\"229\" y=\"171\" width=\"18\" height=\"20\" rx=\"2\" fill=\"#9a4adf\" opacity=\"0.3\"\/>\n <text x=\"229\" y=\"185\" font-size=\"12\" fill=\"#5a1a9a\" font-family=\"Courier New, monospace\" font-weight=\"bold\">2<\/text>\n <text x=\"247\" y=\"185\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">1…<\/text>\n\n <text x=\"20\" y=\"210\" font-size=\"14\" fill=\"#9a4adf\">\u22ee<\/text>\n\n \n <text x=\"20\" y=\"245\" font-size=\"11\" fill=\"#5a1a9a\" font-weight=\"bold\">Chiffres diagonaux : 4, 6, 5, 7, 2, …<\/text>\n <text x=\"20\" y=\"263\" font-size=\"11\" fill=\"#b03a2e\" font-weight=\"bold\">On change chaque chiffre : (\u22604), (\u22606), (\u22605), (\u22607), (\u22602)…<\/text>\n <text x=\"20\" y=\"281\" font-size=\"11\" fill=\"#b03a2e\" font-weight=\"bold\">\u2192 d = 0.57681… est ABSENT de la liste ! Contradiction \u2717<\/text>\n <\/svg>\n <\/div>\n\n <h3>L’argument en clair<\/h3>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> On suppose qu’on a une liste compl\u00e8te de tous les r\u00e9els entre 0 et 1 : r\u2081, r\u2082, r\u2083…<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> On regarde le 1er chiffre de r\u2081, le 2\u00e8me chiffre de r\u2082, le 3\u00e8me chiffre de r\u2083… C’est la <strong>diagonale<\/strong>.<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2462<\/span> On construit un nouveau nombre d en changeant chaque chiffre diagonal : si le chiffre est 4, on met 5 ; sinon on met 4. (N’importe quelle r\u00e8gle qui change le chiffre fonctionne.)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2463<\/span> Ce nombre d est diff\u00e9rent de r\u2081 (car son 1er chiffre diff\u00e8re), diff\u00e9rent de r\u2082 (2\u00e8me chiffre), diff\u00e9rent de r\u2083 (3\u00e8me chiffre)… diff\u00e9rent de <em>chaque<\/em> r\u2099.<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2464<\/span> Donc d n’est pas dans la liste \u2014 <strong>contradiction<\/strong> ! La liste n’\u00e9tait pas compl\u00e8te.<\/div>\n\n <div class=\"choc\">\n <div class=\"choc-titre\">\ud83c\udf0c Conclusion vertigineuse<\/div>\n <div class=\"choc-contenu\">\n Aucune liste ne peut contenir tous les r\u00e9els.<br>\n Les r\u00e9els ne sont pas d\u00e9nombrables.<br>\n Il y a <em>strictement plus<\/em> de r\u00e9els que d’entiers.<br><br>\n <span style=\"font-size:1rem; color:rgba(240,208,96,0.7);\">Le cardinal de \u211d, not\u00e9 \ud835\udd20, est strictement sup\u00e9rieur \u00e0 \u2135\u2080.<\/span>\n <\/div>\n <\/div>\n\n <h2>La hi\u00e9rarchie des infinis<\/h2>\n\n <p>\n Et ce n’est pas tout. Cantor a montr\u00e9 qu’on peut construire des infinis de plus en plus grands, sans jamais s’arr\u00eater. Pour tout ensemble A, l’ensemble de ses parties (not\u00e9 \ud835\udcab(A)) a un cardinal strictement sup\u00e9rieur \u00e0 celui de A.\n <\/p>\n\n <div class=\"formule\">|\ud835\udcab(A)| > |A| \u2014 toujours, m\u00eame si A est infini<\/div>\n\n <p>\n Cela cr\u00e9e une hi\u00e9rarchie infinie d’infinis :\n <\/p>\n\n <div class=\"infinis-scale\">\n <div class=\"infini-row\">\n <div class=\"infini-label\" style=\"color:#3a1a5a; font-size:0.9rem;\">\u2135\u2080<\/div>\n <div class=\"infini-bar\" style=\"width:25%; background:#3a1a5a; font-size:0.8rem;\">\u2115, \u2124, \u211a<\/div>\n <div class=\"infini-desc\">Entiers, relatifs, fractions \u2014 tous pareils<\/div>\n <\/div>\n <div class=\"infini-row\">\n <div class=\"infini-label\" style=\"color:#5a1a9a; font-size:0.9rem;\">\ud835\udd20 = 2^\u2135\u2080<\/div>\n <div class=\"infini-bar\" style=\"width:50%; background:#5a1a9a; font-size:0.8rem;\">\u211d, intervalles, courbes<\/div>\n <div class=\"infini-desc\">Nombres r\u00e9els \u2014 strictement plus grand que \u2135\u2080<\/div>\n <\/div>\n <div class=\"infini-row\">\n <div class=\"infini-label\" style=\"color:#7a2aba; font-size:0.9rem;\">2^\ud835\udd20<\/div>\n <div class=\"infini-bar\" style=\"width:70%; background:#7a2aba; font-size:0.8rem;\">\ud835\udcab(\u211d) \u2014 parties de \u211d<\/div>\n <div class=\"infini-desc\">Encore plus grand \u2014 et on peut continuer…<\/div>\n <\/div>\n <div class=\"infini-row\">\n <div class=\"infini-label\" style=\"color:#9a4adf; font-size:0.9rem;\">2^(2^\ud835\udd20)<\/div>\n <div class=\"infini-bar\" style=\"width:90%; background:#9a4adf; font-size:0.8rem;\">\ud835\udcab(\ud835\udcab(\u211d))…<\/div>\n <div class=\"infini-desc\">Et encore plus… \u00e0 l’infini !<\/div>\n <\/div>\n <\/div>\n\n <h2>L’hypoth\u00e8se du continu \u2014 le myst\u00e8re non r\u00e9solu<\/h2>\n\n <p>\n Cantor a pos\u00e9 une question qui est rest\u00e9e ouverte pendant plus d’un si\u00e8cle : <strong>existe-t-il un infini entre \u2135\u2080 et \ud835\udd20 ?<\/strong> Autrement dit, y a-t-il un niveau d’infini entre celui des entiers et celui des r\u00e9els ?\n <\/p>\n\n <p>\n Cette question s’appelle l’<strong>hypoth\u00e8se du continu<\/strong>. La r\u00e9ponse est l’une des plus troublantes de toute la logique math\u00e9matique : <strong>on ne peut ni la prouver ni la r\u00e9futer<\/strong> dans les axiomes standard des math\u00e9matiques. C’est une question ind\u00e9cidable \u2014 elle d\u00e9passe les limites de notre syst\u00e8me formel.\n <\/p>\n\n <blockquote>\n L’infini est l\u00e0 o\u00f9 les math\u00e9matiques rencontrent la philosophie. Et Cantor a montr\u00e9 que m\u00eame l\u00e0, la logique peut nous guider \u2014 jusqu’\u00e0 une certaine fronti\u00e8re, au-del\u00e0 de laquelle le myst\u00e8re reste entier.\n <cite>\u2014 R\u00e9flexion sur les limites de la connaissance math\u00e9matique<\/cite>\n <\/blockquote>\n\n <h2>Le code Python \u2014 explorer les ensembles infinis<\/h2>\n\n <div class=\"code-block\">\n <span class=\"code-label\">Python<\/span>\n<span class=\"code-comment\"># Explorer les concepts de Cantor avec Python<\/span>\n<span class=\"code-keyword\">import<\/span> random\n<span class=\"code-keyword\">from<\/span> fractions <span class=\"code-keyword\">import<\/span> Fraction\n\n<span class=\"code-comment\"># — D\u00e9monstration 1 : Bijection N \u2192 Pairs —<\/span>\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0Bijection \u2115 \u2192 Nombres pairs (n \u2192 2n) :\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb{‘n’:>4} \u2192 {‘2n’:>6}\u00a0\u00bb<\/span>)\n<span class=\"code-keyword\">for<\/span> n <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">range<\/span>(<span class=\"code-number\">1<\/span>, <span class=\"code-number\">11<\/span>):\n <span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb{n:>4} \u2192 {2*n:>6}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># — D\u00e9monstration 2 : D\u00e9nombrement des rationnels —<\/span>\n<span class=\"code-comment\"># Parcours diagonal du tableau p\/q<\/span>\n<span class=\"code-keyword\">def<\/span> <span class=\"code-function\">rationnels_diagonale<\/span>(limite):\n <span class=\"code-string\">\u00ab\u00a0\u00a0\u00bb\u00a0\u00bbG\u00e9n\u00e8re les fractions positives par diagonale\u00a0\u00bb\u00a0\u00bb\u00a0\u00bb<\/span>\n rationnels = []\n vus = <span class=\"code-function\">set<\/span>()\n diagonale = <span class=\"code-number\">2<\/span>\n <span class=\"code-keyword\">while<\/span> <span class=\"code-function\">len<\/span>(rationnels) < limite:\n <span class=\"code-keyword\">for<\/span> p <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">range<\/span>(<span class=\"code-number\">1<\/span>, diagonale):\n q = diagonale – p\n f = Fraction(p, q)\n <span class=\"code-keyword\">if<\/span> f <span class=\"code-keyword\">not<\/span> <span class=\"code-keyword\">in<\/span> vus:\n vus.<span class=\"code-function\">add<\/span>(f)\n rationnels.<span class=\"code-function\">append<\/span>(f)\n <span class=\"code-keyword\">if<\/span> <span class=\"code-function\">len<\/span>(rationnels) >= limite:\n <span class=\"code-keyword\">break<\/span>\n diagonale += <span class=\"code-number\">1<\/span>\n <span class=\"code-keyword\">return<\/span> rationnels\n\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0\\nD\u00e9nombrement des rationnels par diagonale :\u00a0\u00bb<\/span>)\nrats = <span class=\"code-function\">rationnels_diagonale<\/span>(<span class=\"code-number\">20<\/span>)\n<span class=\"code-keyword\">for<\/span> i, r <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">enumerate<\/span>(rats, <span class=\"code-number\">1<\/span>):\n <span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb r_{i:02d} = {r}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># — D\u00e9monstration 3 : Argument diagonal de Cantor —<\/span>\n<span class=\"code-keyword\">def<\/span> <span class=\"code-function\">cantor_diagonal<\/span>(liste_reels, n_decimales=<span class=\"code-number\">10<\/span>):\n <span class=\"code-string\">\u00ab\u00a0\u00a0\u00bb\u00a0\u00bb\n Construit un r\u00e9el absent de la liste\n par l’argument diagonal de Cantor\n \u00ab\u00a0\u00a0\u00bb\u00a0\u00bb<\/span>\n nombre_diagonal = <span class=\"code-string\">\u00ab\u00a00.\u00a0\u00bb<\/span>\n <span class=\"code-keyword\">for<\/span> i, r <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">enumerate<\/span>(liste_reels[:n_decimales]):\n <span class=\"code-comment\"># Extrait le i-\u00e8me chiffre d\u00e9cimal de r_i<\/span>\n decimales = <span class=\"code-string\">f\u00a0\u00bb{r:.{n_decimales}f}\u00a0\u00bb<\/span>.<span class=\"code-function\">split<\/span>(<span class=\"code-string\">‘.’<\/span>)[<span class=\"code-number\">1<\/span>]\n chiffre = <span class=\"code-function\">int<\/span>(decimales[i]) <span class=\"code-keyword\">if<\/span> i < <span class=\"code-function\">len<\/span>(decimales) <span class=\"code-keyword\">else<\/span> <span class=\"code-number\">0<\/span>\n <span class=\"code-comment\"># R\u00e8gle : si chiffre = 4 \u2192 5, sinon \u2192 4<\/span>\n nouveau = <span class=\"code-string\">‘5’<\/span> <span class=\"code-keyword\">if<\/span> chiffre == <span class=\"code-number\">4<\/span> <span class=\"code-keyword\">else<\/span> <span class=\"code-string\">‘4’<\/span>\n nombre_diagonal += nouveau\n <span class=\"code-keyword\">return<\/span> nombre_diagonal\n\n<span class=\"code-comment\"># Liste \u00ab\u00a0suppos\u00e9e compl\u00e8te\u00a0\u00bb de r\u00e9els al\u00e9atoires<\/span>\nrandom.<span class=\"code-function\">seed<\/span>(<span class=\"code-number\">42<\/span>)\nliste = [random.<span class=\"code-function\">random<\/span>() <span class=\"code-keyword\">for<\/span> _ <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">range<\/span>(<span class=\"code-number\">8<\/span>)]\n\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0\\nArgument diagonal de Cantor :\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0Liste suppos\u00e9e compl\u00e8te :\u00a0\u00bb<\/span>)\n<span class=\"code-keyword\">for<\/span> i, r <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">enumerate<\/span>(liste, <span class=\"code-number\">1<\/span>):\n <span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb r_{i} = {r:.10f}\u00a0\u00bb<\/span>)\n\nabsent = <span class=\"code-function\">cantor_diagonal<\/span>(liste)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb\\nNombre construit par diagonale : {absent}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0\u2192 Ce nombre est absent de la liste par construction !\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0\u2192 Donc aucune liste ne peut \u00eatre compl\u00e8te. CQFD.\u00a0\u00bb<\/span>)\n <\/div>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9 \u2014 du coll\u00e8ge au sup\u00e9rieur<\/h3>\n <p><strong>Niveau coll\u00e8ge :<\/strong> Construis une bijection entre les entiers naturels \u2115 = {0, 1, 2, 3…} et les entiers impairs {1, 3, 5, 7…}. Donne la r\u00e8gle explicite.<\/p>\n <p><strong>Niveau lyc\u00e9e :<\/strong> Montre que l’ensemble \u2124 des entiers relatifs (…-2, -1, 0, 1, 2…) est d\u00e9nombrable en construisant une bijection avec \u2115.<\/p>\n <p><strong>Niveau sup\u00e9rieur :<\/strong> En utilisant l’argument diagonal de Cantor, explique pourquoi l’ensemble de toutes les suites infinies de 0 et de 1 n’est pas d\u00e9nombrable.<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>Niveau coll\u00e8ge :<\/strong><\/p>\n <p>R\u00e8gle : n \u2192 2n + 1<\/p>\n <p>0 \u2192 1, 1 \u2192 3, 2 \u2192 5, 3 \u2192 7, 4 \u2192 9…<\/p>\n <p>C’est bien une bijection : chaque impair vient d’exactement un entier naturel. \u2705<\/p>\n\n <p><strong>Niveau lyc\u00e9e :<\/strong><\/p>\n <p>On liste \u2124 en \u00ab\u00a0zigzaguant\u00a0\u00bb autour de 0 :<\/p>\n <p>0 \u2192 0, 1 \u2192 1, 2 \u2192 -1, 3 \u2192 2, 4 \u2192 -2, 5 \u2192 3, 6 \u2192 -3…<\/p>\n <p>R\u00e8gle formelle : si n est pair \u2192 n\/2 ; si n est impair \u2192 -(n+1)\/2<\/p>\n <p>Chaque entier relatif appara\u00eet exactement une fois \u2192 bijection \u2192 \u2124 est d\u00e9nombrable \u2705<\/p>\n\n <p><strong>Niveau sup\u00e9rieur :<\/strong><\/p>\n <p>Supposons qu’on peut lister toutes les suites : s\u2081, s\u2082, s\u2083… o\u00f9 chaque s\u1d62 est une suite infinie de 0 et 1.<\/p>\n <p>On construit une nouvelle suite d en prenant le i-\u00e8me \u00e9l\u00e9ment de s\u1d62 et en le changeant (0\u21921, 1\u21920).<\/p>\n <p>d diff\u00e8re de s\u2081 en position 1, de s\u2082 en position 2, de s\u2083 en position 3… donc d diff\u00e8re de toute suite de la liste.<\/p>\n <p>d est une suite valide (de 0 et 1) qui n’est pas dans la liste \u2192 contradiction \u2192 l’ensemble n’est pas d\u00e9nombrable. \u2705<\/p>\n <\/div>\n <\/div>\n\n <div class=\"conclusion\">\n <h2>Ce qu’on retient<\/h2>\n <p>\n L’infini n’est pas une montagne unique \u2014 c’est une cha\u00eene de montagnes sans fin. \u2135\u2080, \ud835\udd20, 2^\ud835\udd20… chaque infini est strictement plus grand que le pr\u00e9c\u00e9dent, et il n’y a pas de sommet. Cantor a ouvert une porte que personne n’avait os\u00e9 franchir avant lui \u2014 et derri\u00e8re cette porte, les math\u00e9matiques sont devenues plus vastes, plus profondes, et plus \u00e9tranges que quiconque ne l’avait imagin\u00e9.\n <\/p>\n <p>\n La prochaine fois qu’un \u00e9l\u00e8ve te dit \u00ab\u00a0il y a une infinit\u00e9 de nombres\u00a0\u00bb, tu pourras lui r\u00e9pondre avec le sourire : \u00ab\u00a0Oui \u2014 mais de quelle taille d’infini parles-tu ?\u00a0\u00bb\n <\/p>\n <p>\n Dans le prochain article, nous plongerons dans <strong>les fractales<\/strong> \u2014 ces formes g\u00e9om\u00e9triques infiniment complexes qui se cachent dans les c\u00f4tes d’Afrique, les arbres, les poumons humains, et les march\u00e9s financiers.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>Pourquoi l’infini a plusieurs tailles \u2014 MathsVivantes Philosophie des maths \u00b7 Article #15 Pourquoi l’infinia plusieurs tailles \u2014et certains sont […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[8],"tags":[],"class_list":["post-136","post","type-post","status-publish","format-standard","hentry","category-fondements"],"yoast_head":"\n<title>La taille de l'Infini - AfrikIntelligent Pourquoi l'infini a plusieurs tailles - maths vivants<\/title>\n<meta name=\"description\" content=\"Tous les infinis ne sont pas \u00e9gaux . Cantor a prouv\u00e9 qu'il existe des infinis plus grands . D\u00e9couvrez l'argument diagonal et le vertige des math\u00e9matiques .\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/vmi3217629.contaboserver.net\/?p=136\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"La taille de l'Infini - AfrikIntelligent Pourquoi l'infini a plusieurs tailles - maths vivants\" \/>\n<meta property=\"og:description\" content=\"Tous les infinis ne sont pas \u00e9gaux . Cantor a prouv\u00e9 qu'il existe des infinis plus grands . D\u00e9couvrez l'argument diagonal et le vertige des math\u00e9matiques .\" \/>\n<meta property=\"og:url\" content=\"https:\/\/vmi3217629.contaboserver.net\/?p=136\" \/>\n<meta property=\"og:site_name\" content=\"AfrikIntelligent\" \/>\n<meta property=\"article:published_time\" content=\"2026-04-29T06:15:48+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-04-29T06:15:49+00:00\" \/>\n<meta name=\"author\" content=\"Odilon\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"Odilon\" \/>\n\t<meta name=\"twitter:label2\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data2\" content=\"11 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136\"},\"author\":{\"name\":\"Odilon\",\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/#\\\/schema\\\/person\\\/a3da39574d729b6b3ff7514ccc055923\"},\"headline\":\"La taille de l’Infini\",\"datePublished\":\"2026-04-29T06:15:48+00:00\",\"dateModified\":\"2026-04-29T06:15:49+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136\"},\"wordCount\":2062,\"commentCount\":0,\"articleSection\":[\"Fondements\"],\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136\",\"url\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136\",\"name\":\"La taille de l'Infini - AfrikIntelligent Pourquoi l'infini a plusieurs tailles - maths vivants\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/#website\"},\"datePublished\":\"2026-04-29T06:15:48+00:00\",\"dateModified\":\"2026-04-29T06:15:49+00:00\",\"author\":{\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/#\\\/schema\\\/person\\\/a3da39574d729b6b3ff7514ccc055923\"},\"description\":\"Tous les infinis ne sont pas \u00e9gaux . Cantor a prouv\u00e9 qu'il existe des infinis plus grands . D\u00e9couvrez l'argument diagonal et le vertige des math\u00e9matiques .\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136#breadcrumb\"},\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Accueil\",\"item\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"La taille de l’Infini\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/#website\",\"url\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/\",\"name\":\"AfrikIntelligent\",\"description\":\"Append autrement\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"fr-FR\"},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/#\\\/schema\\\/person\\\/a3da39574d729b6b3ff7514ccc055923\",\"name\":\"Odilon\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/57bcd6a76fa395ae5e994943b66ccd3dd7e71cd0b1b77b351986e88c27801523?s=96&d=mm&r=g\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/57bcd6a76fa395ae5e994943b66ccd3dd7e71cd0b1b77b351986e88c27801523?s=96&d=mm&r=g\",\"contentUrl\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/57bcd6a76fa395ae5e994943b66ccd3dd7e71cd0b1b77b351986e88c27801523?s=96&d=mm&r=g\",\"caption\":\"Odilon\"},\"sameAs\":[\"https:\\\/\\\/vmi3217629.contaboserver.net\"],\"url\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?author=1\"}]}<\/script>\n","yoast_head_json":{"title":"La taille de l'Infini - AfrikIntelligent Pourquoi l'infini a plusieurs tailles - maths vivants","description":"Tous les infinis ne sont pas \u00e9gaux . Cantor a prouv\u00e9 qu'il existe des infinis plus grands . D\u00e9couvrez l'argument diagonal et le vertige des math\u00e9matiques .","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/vmi3217629.contaboserver.net\/?p=136","og_locale":"fr_FR","og_type":"article","og_title":"La taille de l'Infini - AfrikIntelligent Pourquoi l'infini a plusieurs tailles - maths vivants","og_description":"Tous les infinis ne sont pas \u00e9gaux . Cantor a prouv\u00e9 qu'il existe des infinis plus grands . D\u00e9couvrez l'argument diagonal et le vertige des math\u00e9matiques .","og_url":"https:\/\/vmi3217629.contaboserver.net\/?p=136","og_site_name":"AfrikIntelligent","article_published_time":"2026-04-29T06:15:48+00:00","article_modified_time":"2026-04-29T06:15:49+00:00","author":"Odilon","twitter_card":"summary_large_image","twitter_misc":{"\u00c9crit par":"Odilon","Dur\u00e9e de lecture estim\u00e9e":"11 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/vmi3217629.contaboserver.net\/?p=136#article","isPartOf":{"@id":"https:\/\/vmi3217629.contaboserver.net\/?p=136"},"author":{"name":"Odilon","@id":"https:\/\/vmi3217629.contaboserver.net\/#\/schema\/person\/a3da39574d729b6b3ff7514ccc055923"},"headline":"La taille de l’Infini","datePublished":"2026-04-29T06:15:48+00:00","dateModified":"2026-04-29T06:15:49+00:00","mainEntityOfPage":{"@id":"https:\/\/vmi3217629.contaboserver.net\/?p=136"},"wordCount":2062,"commentCount":0,"articleSection":["Fondements"],"inLanguage":"fr-FR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/vmi3217629.contaboserver.net\/?p=136#respond"]}]},{"@type":"WebPage","@id":"https:\/\/vmi3217629.contaboserver.net\/?p=136","url":"https:\/\/vmi3217629.contaboserver.net\/?p=136","name":"La taille de l'Infini - AfrikIntelligent Pourquoi l'infini a plusieurs tailles - maths vivants","isPartOf":{"@id":"https:\/\/vmi3217629.contaboserver.net\/#website"},"datePublished":"2026-04-29T06:15:48+00:00","dateModified":"2026-04-29T06:15:49+00:00","author":{"@id":"https:\/\/vmi3217629.contaboserver.net\/#\/schema\/person\/a3da39574d729b6b3ff7514ccc055923"},"description":"Tous les infinis ne sont pas \u00e9gaux . Cantor a prouv\u00e9 qu'il existe des infinis plus grands . D\u00e9couvrez l'argument diagonal et le vertige des math\u00e9matiques .","breadcrumb":{"@id":"https:\/\/vmi3217629.contaboserver.net\/?p=136#breadcrumb"},"inLanguage":"fr-FR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/vmi3217629.contaboserver.net\/?p=136"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/vmi3217629.contaboserver.net\/?p=136#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Accueil","item":"https:\/\/vmi3217629.contaboserver.net\/"},{"@type":"ListItem","position":2,"name":"La taille de l’Infini"}]},{"@type":"WebSite","@id":"https:\/\/vmi3217629.contaboserver.net\/#website","url":"https:\/\/vmi3217629.contaboserver.net\/","name":"AfrikIntelligent","description":"Append autrement","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/vmi3217629.contaboserver.net\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"fr-FR"},{"@type":"Person","@id":"https:\/\/vmi3217629.contaboserver.net\/#\/schema\/person\/a3da39574d729b6b3ff7514ccc055923","name":"Odilon","image":{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/secure.gravatar.com\/avatar\/57bcd6a76fa395ae5e994943b66ccd3dd7e71cd0b1b77b351986e88c27801523?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/57bcd6a76fa395ae5e994943b66ccd3dd7e71cd0b1b77b351986e88c27801523?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/57bcd6a76fa395ae5e994943b66ccd3dd7e71cd0b1b77b351986e88c27801523?s=96&d=mm&r=g","caption":"Odilon"},"sameAs":["https:\/\/vmi3217629.contaboserver.net"],"url":"https:\/\/vmi3217629.contaboserver.net\/?author=1"}]}},"uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false},"uagb_author_info":{"display_name":"Odilon","author_link":"https:\/\/vmi3217629.contaboserver.net\/?author=1"},"uagb_comment_info":0,"uagb_excerpt":"Pourquoi l’infini a plusieurs tailles \u2014 MathsVivantes Philosophie des maths \u00b7 Article #15 Pourquoi l’infinia plusieurs tailles \u2014et certains sont […]","_links":{"self":[{"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/posts\/136","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=136"}],"version-history":[{"count":1,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/posts\/136\/revisions"}],"predecessor-version":[{"id":137,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/posts\/136\/revisions\/137"}],"wp:attachment":[{"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=136"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=136"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=136"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}

{"id":136,"date":"2026-04-29T07:15:48","date_gmt":"2026-04-29T06:15:48","guid":{"rendered":"https:\/\/vmi3217629.contaboserver.net\/?p=136"},"modified":"2026-04-29T07:15:49","modified_gmt":"2026-04-29T06:15:49","slug":"infini-plusieurs-tailles-cantor","status":"publish","type":"post","link":"https:\/\/vmi3217629.contaboserver.net\/?p=136","title":{"rendered":"La taille de l’Infini"},"content":{"rendered":"\n\n\n\n\n\nPourquoi l’infini a plusieurs tailles \u2014 MathsVivantes<\/title>\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Playfair+Display:ital,wght@0,400;0,700;1,400&family=Source+Serif+4:ital,wght@0,300;0,400;0,600;1,300;1,400&display=swap\" rel=\"stylesheet\">\n<style>\n :root {\n --noir: #0f0e0b;\n --creme: #f5f0e8;\n --cosmos: #0a0520;\n --violet: #5a1a9a;\n --violet-clair: #9a4adf;\n --violet-pale: #f5eeff;\n --or: #c8a010;\n --or-clair: #f0d060;\n --rouge: #b03a2e;\n --gris: #6b6560;\n --gris-clair: #e8e2d8;\n }\n\n * { margin: 0; padding: 0; box-sizing: border-box; }\n\n body {\n background-color: var(--creme);\n color: var(--noir);\n font-family: 'Source Serif 4', Georgia, serif;\n font-weight: 300;\n line-height: 1.8;\n }\n\n .hero {\n background: #04010f;\n color: var(--creme);\n padding: 80px 40px 70px;\n position: relative;\n overflow: hidden;\n }\n .hero::before {\n content: \"\u221e \u2135\u2080 \u2135\u2081 \u221e\";\n position: absolute;\n top: 20px; right: 30px;\n font-size: 5rem;\n opacity: 0.05;\n font-family: 'Playfair Display', serif;\n letter-spacing: 0.3em;\n }\n .hero-inner { max-width: 760px; margin: 0 auto; }\n\n .article-rubrique {\n display: inline-block;\n background: var(--violet);\n color: white;\n font-size: 0.7rem;\n letter-spacing: 0.18em;\n text-transform: uppercase;\n padding: 4px 12px;\n margin-bottom: 24px;\n font-weight: 600;\n }\n\n .hero h1 {\n font-family: 'Playfair Display', serif;\n font-size: clamp(2rem, 5vw, 3.2rem);\n line-height: 1.2;\n margin-bottom: 24px;\n font-weight: 700;\n }\n .hero h1 em { color: var(--or-clair); font-style: italic; }\n\n .hero-intro {\n font-size: 1.15rem;\n opacity: 0.75;\n max-width: 600px;\n font-style: italic;\n border-left: 3px solid var(--violet-clair);\n padding-left: 20px;\n margin-bottom: 32px;\n }\n .meta { font-size: 0.8rem; opacity: 0.45; letter-spacing: 0.08em; text-transform: uppercase; }\n\n .article-body { max-width: 760px; margin: 0 auto; padding: 60px 40px 80px; }\n .article-body p { font-size: 1.05rem; margin-bottom: 1.6em; color: #2a2520; }\n\n h2 {\n font-family: 'Playfair Display', serif;\n font-size: 1.6rem;\n margin: 2.5em 0 0.8em;\n color: var(--noir);\n font-weight: 700;\n }\n h2::before { content: \"\u00a7 \"; color: var(--violet-clair); }\n\n h3 {\n font-family: 'Playfair Display', serif;\n font-size: 1.15rem;\n font-weight: 700;\n margin: 1.8em 0 0.6em;\n color: var(--violet);\n }\n\n .encadre {\n border: 1.5px solid var(--violet-clair);\n background: white;\n padding: 24px 28px;\n margin: 2em 0;\n position: relative;\n }\n .encadre-titre {\n position: absolute;\n top: -12px; left: 20px;\n background: white;\n padding: 0 8px;\n font-size: 0.72rem;\n letter-spacing: 0.15em;\n text-transform: uppercase;\n color: var(--violet-clair);\n font-weight: 600;\n }\n .encadre p { margin-bottom: 0.8em; font-size: 0.98rem; }\n .encadre p:last-child { margin-bottom: 0; }\n\n .encadre-histoire {\n background: #04010f;\n color: var(--creme);\n border: none;\n padding: 32px 36px;\n margin: 2.5em 0;\n }\n .encadre-histoire .encadre-titre { background: #04010f; color: var(--or-clair); }\n .encadre-histoire p { color: rgba(245,240,232,0.85); }\n\n .definition {\n background: var(--violet-pale);\n border-left: 4px solid var(--violet-clair);\n padding: 20px 24px;\n margin: 1.8em 0;\n font-style: italic;\n }\n .definition strong { font-style: normal; color: var(--violet); font-family: 'Playfair Display', serif; }\n\n .choc {\n background: #04010f;\n color: var(--creme);\n padding: 28px 32px;\n margin: 2em 0;\n text-align: center;\n }\n .choc-titre {\n font-size: 0.72rem;\n letter-spacing: 0.18em;\n text-transform: uppercase;\n color: var(--violet-clair);\n margin-bottom: 12px;\n font-weight: 600;\n }\n .choc-contenu {\n font-family: 'Playfair Display', serif;\n font-size: 1.3rem;\n color: var(--or-clair);\n line-height: 1.6;\n }\n\n .schema-container {\n background: white;\n border: 1.5px solid var(--violet-clair);\n padding: 24px;\n margin: 2em 0;\n text-align: center;\n }\n .schema-titre {\n font-size: 0.72rem;\n letter-spacing: 0.15em;\n text-transform: uppercase;\n color: var(--violet-clair);\n font-weight: 600;\n margin-bottom: 16px;\n }\n\n .formule {\n text-align: center;\n font-size: 1.25rem;\n font-family: 'Playfair Display', serif;\n margin: 1.5em 0;\n color: var(--noir);\n padding: 16px;\n background: white;\n border-top: 2px solid var(--violet-clair);\n border-bottom: 2px solid var(--violet-clair);\n }\n\n .etape {\n background: white;\n border-left: 3px solid var(--violet-clair);\n padding: 14px 20px;\n margin: 0.6em 0;\n font-size: 0.97rem;\n }\n .etape-numero {\n font-family: 'Playfair Display', serif;\n font-weight: 700;\n color: var(--violet-clair);\n margin-right: 8px;\n }\n\n \/* HI\u00c9RARCHIE DES INFINIS *\/\n .infinis-scale {\n margin: 2em 0;\n display: flex;\n flex-direction: column;\n gap: 8px;\n }\n .infini-row {\n display: flex;\n align-items: center;\n gap: 16px;\n }\n .infini-label {\n min-width: 80px;\n font-family: 'Playfair Display', serif;\n font-size: 1.1rem;\n font-weight: 700;\n text-align: right;\n }\n .infini-bar {\n height: 36px;\n display: flex;\n align-items: center;\n padding: 0 16px;\n font-size: 0.85rem;\n color: white;\n font-weight: 600;\n border-radius: 2px;\n }\n .infini-desc {\n font-size: 0.85rem;\n color: var(--gris);\n font-style: italic;\n }\n\n .code-block {\n background: #04010f;\n color: #d0b0f0;\n padding: 24px 28px;\n margin: 2em 0;\n font-family: 'Courier New', monospace;\n font-size: 0.88rem;\n line-height: 1.7;\n overflow-x: auto;\n position: relative;\n }\n .code-label { position: absolute; top: 8px; right: 12px; font-size: 0.65rem; letter-spacing: 0.15em; text-transform: uppercase; color: var(--violet-clair); font-family: 'Source Serif 4', serif; }\n .code-comment { color: #5a3a8a; }\n .code-keyword { color: #a080d0; }\n .code-string { color: var(--or-clair); }\n .code-number { color: #c0a0f0; }\n .code-function { color: #e0c0ff; }\n\n .exercice {\n background: #04010f;\n color: var(--creme);\n padding: 36px 40px;\n margin: 3em 0;\n }\n .exercice h3 { color: var(--or-clair); font-style: normal; margin-top: 0; margin-bottom: 1em; font-size: 1.1rem; font-weight: 600; }\n .exercice p { color: rgba(245,240,232,0.82); font-size: 0.97rem; margin-bottom: 0.8em; }\n .exercice .correction { border-top: 1px solid rgba(90,26,154,0.4); margin-top: 1.5em; padding-top: 1.2em; }\n .exercice .correction-titre { font-size: 0.7rem; letter-spacing: 0.18em; text-transform: uppercase; color: var(--or-clair); margin-bottom: 0.8em; font-weight: 600; }\n\n blockquote {\n padding: 0 0 0 32px;\n margin: 2em 0;\n position: relative;\n font-style: italic;\n font-size: 1.12rem;\n color: #3a3530;\n }\n blockquote::before {\n content: \"\\201C\";\n position: absolute;\n left: 0; top: -10px;\n font-size: 3.5rem;\n color: var(--violet-clair);\n font-family: 'Playfair Display', serif;\n line-height: 1;\n }\n blockquote cite { display: block; font-size: 0.82rem; font-style: normal; letter-spacing: 0.1em; text-transform: uppercase; color: var(--gris); margin-top: 8px; }\n\n .conclusion { border-top: 2px solid var(--noir); margin-top: 3em; padding-top: 2em; }\n\n @media (max-width: 600px) {\n .hero { padding: 50px 20px 40px; }\n .article-body { padding: 40px 20px 60px; }\n .exercice { padding: 24px 20px; }\n .code-block { padding: 16px; font-size: 0.78rem; }\n .infini-row { flex-wrap: wrap; }\n }\n<\/style>\n<\/head>\n<body>\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">Philosophie des maths \u00b7 Article #15<\/div>\n <h1>Pourquoi l’infini<br>a <em>plusieurs tailles<\/em> \u2014<br>et certains sont plus grands<\/h1>\n <p class=\"hero-intro\">Combien y a-t-il de nombres entiers ? Infini. Combien y a-t-il de nombres r\u00e9els ? Aussi infini. Mais ces deux infinis ne sont pas \u00e9gaux. L’un est strictement plus grand que l’autre. Et c’est d\u00e9montrable. Bienvenue dans l’une des id\u00e9es les plus vertigineuses de toute l’histoire des math\u00e9matiques.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 13 min<\/div>\n <\/div>\n<\/div>\n\n<article class=\"article-body\">\n\n <p>\n Imagine que tu comptes les grains de sable de la plage de Fidjross\u00e8 \u00e0 Cotonou. Tu n’y arriveras jamais \u2014 il y en a trop. Maintenant imagine que tu comptes les \u00e9toiles de l’univers. Encore plus impossible. Ces deux quantit\u00e9s sont toutes les deux \u00ab\u00a0infinies\u00a0\u00bb dans le sens pratique \u2014 mais sont-elles <em>\u00e9gales<\/em> ?\n <\/p>\n\n <p>\n Un math\u00e9maticien allemand nomm\u00e9 Georg Cantor a pos\u00e9 cette question au XIXe si\u00e8cle et a obtenu une r\u00e9ponse qui a failli le rendre fou \u2014 et qui a choqu\u00e9 la communaut\u00e9 math\u00e9matique mondiale. Sa r\u00e9ponse : <strong>non, tous les infinis ne sont pas \u00e9gaux. Certains infinis sont strictement plus grands que d’autres.<\/strong>\n <\/p>\n\n <p>\n Et il l’a <em>prouv\u00e9<\/em>.\n <\/p>\n\n <h2>Commen\u00e7ons par le commencement : compter sans nombres<\/h2>\n\n <p>\n Avant d’attaquer l’infini, il faut r\u00e9apprendre ce que signifie \u00ab\u00a0m\u00eame taille\u00a0\u00bb. La d\u00e9finition intuitive \u2014 compter les \u00e9l\u00e9ments et comparer \u2014 ne fonctionne pas pour des ensembles infinis. Il faut une d\u00e9finition plus profonde.\n <\/p>\n\n <div class=\"definition\">\n <strong>Bijection (ou correspondance biunivoque) :<\/strong> Une bijection entre deux ensembles A et B est une r\u00e8gle qui associe \u00e0 chaque \u00e9l\u00e9ment de A exactement un \u00e9l\u00e9ment de B, et vice versa \u2014 sans rien oublier, sans rien r\u00e9p\u00e9ter. Si une telle bijection existe, les deux ensembles ont la <em>m\u00eame taille<\/em> \u2014 le m\u00eame <strong>cardinal<\/strong>.\n <\/div>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83c\udf0d Analogie du march\u00e9 de Dantokpa<\/span>\n <p>Imagine deux files au march\u00e9 : une file de vendeuses et une file d’acheteurs. Pour savoir si les deux files ont le m\u00eame nombre de personnes, tu n’as pas besoin de les compter. Il suffit de les <strong>jumeler<\/strong> \u2014 chaque vendeuse face \u00e0 un acheteur. S’il n’y a ni vendeuse ni acheteur sans partenaire \u00e0 la fin, les deux files ont le m\u00eame effectif.<\/p>\n <p>C’est exactement la d\u00e9finition d’une bijection. Et c’est le seul outil rigoureux pour comparer des ensembles infinis.<\/p>\n <\/div>\n\n <h2>Premier choc : les nombres pairs sont \u00ab\u00a0aussi nombreux\u00a0\u00bb que les entiers<\/h2>\n\n <p>\n Voici quelque chose d’imm\u00e9diatement contre-intuitif. Les nombres pairs (2, 4, 6, 8…) semblent deux fois moins nombreux que les entiers (1, 2, 3, 4…). Pourtant, en math\u00e9matiques rigoureuses, ils ont le <em>m\u00eame cardinal<\/em>.\n <\/p>\n\n <p>Preuve : construisons une bijection entre \u2115 (entiers naturels) et les nombres pairs :<\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udd17 Bijection entre entiers et nombres pairs<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 160\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"160\" fill=\"white\"\/>\n\n \n <text x=\"30\" y=\"40\" font-size=\"12\" fill=\"#5a1a9a\" font-weight=\"bold\">\u2115 :<\/text>\n <rect x=\"60\" y=\"22\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#f5eeff\" stroke=\"#9a4adf\" stroke-width=\"1.5\"\/>\n <text x=\"85\" y=\"41\" font-size=\"13\" fill=\"#5a1a9a\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">1<\/text>\n <rect x=\"120\" y=\"22\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#f5eeff\" stroke=\"#9a4adf\" stroke-width=\"1.5\"\/>\n <text x=\"145\" y=\"41\" font-size=\"13\" fill=\"#5a1a9a\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">2<\/text>\n <rect x=\"180\" y=\"22\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#f5eeff\" stroke=\"#9a4adf\" stroke-width=\"1.5\"\/>\n <text x=\"205\" y=\"41\" font-size=\"13\" fill=\"#5a1a9a\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">3<\/text>\n <rect x=\"240\" y=\"22\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#f5eeff\" stroke=\"#9a4adf\" stroke-width=\"1.5\"\/>\n <text x=\"265\" y=\"41\" font-size=\"13\" fill=\"#5a1a9a\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">4<\/text>\n <rect x=\"300\" y=\"22\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#f5eeff\" stroke=\"#9a4adf\" stroke-width=\"1.5\"\/>\n <text x=\"325\" y=\"41\" font-size=\"13\" fill=\"#5a1a9a\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">5<\/text>\n <text x=\"370\" y=\"41\" font-size=\"16\" fill=\"#9a4adf\">\u2192 \u221e<\/text>\n\n \n <line x1=\"85\" y1=\"50\" x2=\"85\" y2=\"88\" stroke=\"#9a4adf\" stroke-width=\"1.5\" marker-end=\"url(#arrV)\"\/>\n <line x1=\"145\" y1=\"50\" x2=\"145\" y2=\"88\" stroke=\"#9a4adf\" stroke-width=\"1.5\" marker-end=\"url(#arrV)\"\/>\n <line x1=\"205\" y1=\"50\" x2=\"205\" y2=\"88\" stroke=\"#9a4adf\" stroke-width=\"1.5\" marker-end=\"url(#arrV)\"\/>\n <line x1=\"265\" y1=\"50\" x2=\"265\" y2=\"88\" stroke=\"#9a4adf\" stroke-width=\"1.5\" marker-end=\"url(#arrV)\"\/>\n <line x1=\"325\" y1=\"50\" x2=\"325\" y2=\"88\" stroke=\"#9a4adf\" stroke-width=\"1.5\" marker-end=\"url(#arrV)\"\/>\n <defs>\n <marker id=\"arrV\" markerWidth=\"7\" markerHeight=\"7\" refX=\"5\" refY=\"3\" orient=\"auto\">\n <path d=\"M0,0 L0,6 L7,3 z\" fill=\"#9a4adf\"\/>\n <\/marker>\n <\/defs>\n\n \n <text x=\"440\" y=\"72\" font-size=\"11\" fill=\"#9a4adf\" font-style=\"italic\">n \u2192 2n<\/text>\n\n \n <text x=\"30\" y=\"118\" font-size=\"12\" fill=\"#5a1a9a\" font-weight=\"bold\">Pairs :<\/text>\n <rect x=\"60\" y=\"100\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#5a1a9a\"\/>\n <text x=\"85\" y=\"119\" font-size=\"13\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">2<\/text>\n <rect x=\"120\" y=\"100\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#5a1a9a\"\/>\n <text x=\"145\" y=\"119\" font-size=\"13\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">4<\/text>\n <rect x=\"180\" y=\"100\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#5a1a9a\"\/>\n <text x=\"205\" y=\"119\" font-size=\"13\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">6<\/text>\n <rect x=\"240\" y=\"100\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#5a1a9a\"\/>\n <text x=\"265\" y=\"119\" font-size=\"13\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">8<\/text>\n <rect x=\"300\" y=\"100\" width=\"50\" height=\"28\" rx=\"3\" fill=\"#5a1a9a\"\/>\n <text x=\"325\" y=\"119\" font-size=\"13\" fill=\"white\" text-anchor=\"middle\" font-family=\"Playfair Display, serif\">10<\/text>\n <text x=\"370\" y=\"119\" font-size=\"16\" fill=\"#5a1a9a\">\u2192 \u221e<\/text>\n\n \n <text x=\"260\" y=\"150\" font-size=\"11\" fill=\"#5a1a9a\" text-anchor=\"middle\">Bijection parfaite : m\u00eame cardinal !<\/text>\n <\/svg>\n <\/div>\n\n <p>\n La r\u00e8gle n \u2192 2n est une bijection parfaite : chaque entier correspond exactement \u00e0 un pair, et chaque pair vient d’exactement un entier. Donc les entiers et les pairs ont le <strong>m\u00eame cardinal<\/strong> \u2014 m\u00eame si les pairs semblent \u00ab\u00a0moins nombreux\u00a0\u00bb.\n <\/p>\n\n <p>\n C’est le premier signe que l’infini ne se comporte pas comme les nombres finis. Avec l’infini, une partie peut \u00eatre aussi grande que le tout. C’est contre-intuitif \u2014 et c’est rigoureusement vrai.\n <\/p>\n\n <h2>Les infinis d\u00e9nombrables \u2014 le premier niveau<\/h2>\n\n <div class=\"definition\">\n <strong>Ensemble d\u00e9nombrable :<\/strong> Un ensemble dont les \u00e9l\u00e9ments peuvent \u00eatre mis en bijection avec les entiers naturels \u2115. Autrement dit, on peut les \u00ab\u00a0lister\u00a0\u00bb \u2014 m\u00eame si la liste est infinie. Le cardinal de \u2115 se note \u2135\u2080 (aleph-z\u00e9ro).\n <\/div>\n\n <p>\n Voici des ensembles qui sont tous d\u00e9nombrables \u2014 ils ont tous le m\u00eame cardinal \u2135\u2080 :\n <\/p>\n\n <div class=\"infinis-scale\">\n <div class=\"infini-row\">\n <div class=\"infini-label\" style=\"color:#5a1a9a;\">\u2115<\/div>\n <div class=\"infini-bar\" style=\"width:80%; background:#5a1a9a;\">0, 1, 2, 3, 4, 5, 6…<\/div>\n <div class=\"infini-desc\">Entiers naturels \u2014 la d\u00e9finition<\/div>\n <\/div>\n <div class=\"infini-row\">\n <div class=\"infini-label\" style=\"color:#7a2aba;\">\u2124<\/div>\n <div class=\"infini-bar\" style=\"width:80%; background:#7a2aba;\">…-2, -1, 0, 1, 2…<\/div>\n <div class=\"infini-desc\">Entiers relatifs \u2014 aussi d\u00e9nombrables !<\/div>\n <\/div>\n <div class=\"infini-row\">\n <div class=\"infini-label\" style=\"color:#9a4adf;\">\u211a<\/div>\n <div class=\"infini-bar\" style=\"width:80%; background:#9a4adf;\">1\/2, 3\/4, 22\/7… toutes les fractions<\/div>\n <div class=\"infini-desc\">Rationnels \u2014 aussi d\u00e9nombrables ! (Cantor l’a prouv\u00e9)<\/div>\n <\/div>\n <\/div>\n\n <p>\n Oui \u2014 il y a autant de fractions que d’entiers naturels. Cantor a d\u00e9montr\u00e9 cela avec une astuce g\u00e9niale : parcourir les fractions en diagonale dans un tableau infini. C’est ce qu’on appelle le <strong>d\u00e9nombrement diagonal<\/strong>.\n <\/p>\n\n <h2>Deuxi\u00e8me choc : les r\u00e9els ne sont PAS d\u00e9nombrables<\/h2>\n\n <p>\n Maintenant, qu’en est-il des nombres r\u00e9els \u211d \u2014 tous les nombres de la droite num\u00e9rique, rationnels et irrationnels confondus ? Sont-ils d\u00e9nombrables ?\n <\/p>\n\n <p>\n Cantor a r\u00e9pondu avec une d\u00e9monstration d’une beaut\u00e9 et d’une simplicit\u00e9 d\u00e9vastatrices. C’est la <strong>diagonale de Cantor<\/strong> \u2014 consid\u00e9r\u00e9e comme l’une des plus belles preuves de toute l’histoire des math\u00e9matiques.\n <\/p>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83e\udde0 Georg Cantor \u2014 l’homme qui a os\u00e9 compter l’infini<\/span>\n <p>\n Georg Cantor (1845\u20131918) \u00e9tait un math\u00e9maticien russo-allemand. Son travail sur les ensembles infinis a \u00e9t\u00e9 r\u00e9volutionnaire \u2014 et sa vie, tragique. Ses id\u00e9es ont \u00e9t\u00e9 violemment rejet\u00e9es par les math\u00e9maticiens de son \u00e9poque, notamment son ancien professeur Leopold Kronecker qui disait de lui qu’il \u00e9tait un \u00ab\u00a0corrupteur de la jeunesse\u00a0\u00bb.\n <\/p>\n <p>\n Cantor a souffert de d\u00e9pression s\u00e9v\u00e8re toute sa vie \u2014 en partie \u00e0 cause du rejet de ses travaux. Il est mort dans un sanatorium en 1918. Moins de 20 ans plus tard, ses id\u00e9es sur l’infini sont devenues les fondements de toute la math\u00e9matique moderne. David Hilbert, le plus grand math\u00e9maticien de l’\u00e9poque, a dit : <em>\u00ab\u00a0Personne ne nous chassera du paradis que Cantor a cr\u00e9\u00e9.\u00a0\u00bb<\/em>\n <\/p>\n <p>\n Ce n’est pas la premi\u00e8re fois dans l’histoire qu’un g\u00e9nie incompris par ses contemporains a d\u00fb attendre la post\u00e9rit\u00e9 pour \u00eatre reconnu. Galil\u00e9e, Boltzmann, Abel \u2014 la liste est longue.\n <\/p>\n <\/div>\n\n <h2>L’argument diagonal de Cantor \u2014 la preuve compl\u00e8te<\/h2>\n\n <p>\n Nous allons prouver que les nombres r\u00e9els entre 0 et 1 ne sont <em>pas<\/em> d\u00e9nombrables. La m\u00e9thode : on suppose qu’ils le sont, et on montre une contradiction.\n <\/p>\n\n <p>\n <strong>Hypoth\u00e8se :<\/strong> Supposons qu’on peut lister tous les r\u00e9els entre 0 et 1 \u2014 une liste infinie mais compl\u00e8te. Appelons-les r\u2081, r\u2082, r\u2083, r\u2084…\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcd0 L’argument diagonal de Cantor<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 300\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"300\" fill=\"white\"\/>\n\n \n <text x=\"20\" y=\"35\" font-size=\"12\" fill=\"#5a1a9a\" font-weight=\"bold\" font-family=\"Playfair Display, serif\">La liste suppos\u00e9e compl\u00e8te :<\/text>\n\n \n <text x=\"20\" y=\"65\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">r\u2081 = 0.<\/text>\n <rect x=\"85\" y=\"51\" width=\"18\" height=\"20\" rx=\"2\" fill=\"#9a4adf\" opacity=\"0.3\"\/>\n <text x=\"85\" y=\"65\" font-size=\"12\" fill=\"#5a1a9a\" font-family=\"Courier New, monospace\" font-weight=\"bold\">4<\/text>\n <text x=\"103\" y=\"65\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">1 5 9 2 6…<\/text>\n\n \n <text x=\"20\" y=\"95\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">r\u2082 = 0.1<\/text>\n <rect x=\"121\" y=\"81\" width=\"18\" height=\"20\" rx=\"2\" fill=\"#9a4adf\" opacity=\"0.3\"\/>\n <text x=\"121\" y=\"95\" font-size=\"12\" fill=\"#5a1a9a\" font-family=\"Courier New, monospace\" font-weight=\"bold\">6<\/text>\n <text x=\"139\" y=\"95\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">2 1 8 2…<\/text>\n\n \n <text x=\"20\" y=\"125\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">r\u2083 = 0.26<\/text>\n <rect x=\"157\" y=\"111\" width=\"18\" height=\"20\" rx=\"2\" fill=\"#9a4adf\" opacity=\"0.3\"\/>\n <text x=\"157\" y=\"125\" font-size=\"12\" fill=\"#5a1a9a\" font-family=\"Courier New, monospace\" font-weight=\"bold\">5<\/text>\n <text x=\"175\" y=\"125\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">3 5 8…<\/text>\n\n \n <text x=\"20\" y=\"155\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">r\u2084 = 0.589<\/text>\n <rect x=\"193\" y=\"141\" width=\"18\" height=\"20\" rx=\"2\" fill=\"#9a4adf\" opacity=\"0.3\"\/>\n <text x=\"193\" y=\"155\" font-size=\"12\" fill=\"#5a1a9a\" font-family=\"Courier New, monospace\" font-weight=\"bold\">7<\/text>\n <text x=\"211\" y=\"155\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">3 2…<\/text>\n\n \n <text x=\"20\" y=\"185\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">r\u2085 = 0.7314<\/text>\n <rect x=\"229\" y=\"171\" width=\"18\" height=\"20\" rx=\"2\" fill=\"#9a4adf\" opacity=\"0.3\"\/>\n <text x=\"229\" y=\"185\" font-size=\"12\" fill=\"#5a1a9a\" font-family=\"Courier New, monospace\" font-weight=\"bold\">2<\/text>\n <text x=\"247\" y=\"185\" font-size=\"12\" fill=\"#3a3530\" font-family=\"Courier New, monospace\">1…<\/text>\n\n <text x=\"20\" y=\"210\" font-size=\"14\" fill=\"#9a4adf\">\u22ee<\/text>\n\n \n <text x=\"20\" y=\"245\" font-size=\"11\" fill=\"#5a1a9a\" font-weight=\"bold\">Chiffres diagonaux : 4, 6, 5, 7, 2, …<\/text>\n <text x=\"20\" y=\"263\" font-size=\"11\" fill=\"#b03a2e\" font-weight=\"bold\">On change chaque chiffre : (\u22604), (\u22606), (\u22605), (\u22607), (\u22602)…<\/text>\n <text x=\"20\" y=\"281\" font-size=\"11\" fill=\"#b03a2e\" font-weight=\"bold\">\u2192 d = 0.57681… est ABSENT de la liste ! Contradiction \u2717<\/text>\n <\/svg>\n <\/div>\n\n <h3>L’argument en clair<\/h3>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> On suppose qu’on a une liste compl\u00e8te de tous les r\u00e9els entre 0 et 1 : r\u2081, r\u2082, r\u2083…<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> On regarde le 1er chiffre de r\u2081, le 2\u00e8me chiffre de r\u2082, le 3\u00e8me chiffre de r\u2083… C’est la <strong>diagonale<\/strong>.<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2462<\/span> On construit un nouveau nombre d en changeant chaque chiffre diagonal : si le chiffre est 4, on met 5 ; sinon on met 4. (N’importe quelle r\u00e8gle qui change le chiffre fonctionne.)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2463<\/span> Ce nombre d est diff\u00e9rent de r\u2081 (car son 1er chiffre diff\u00e8re), diff\u00e9rent de r\u2082 (2\u00e8me chiffre), diff\u00e9rent de r\u2083 (3\u00e8me chiffre)… diff\u00e9rent de <em>chaque<\/em> r\u2099.<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2464<\/span> Donc d n’est pas dans la liste \u2014 <strong>contradiction<\/strong> ! La liste n’\u00e9tait pas compl\u00e8te.<\/div>\n\n <div class=\"choc\">\n <div class=\"choc-titre\">\ud83c\udf0c Conclusion vertigineuse<\/div>\n <div class=\"choc-contenu\">\n Aucune liste ne peut contenir tous les r\u00e9els.<br>\n Les r\u00e9els ne sont pas d\u00e9nombrables.<br>\n Il y a <em>strictement plus<\/em> de r\u00e9els que d’entiers.<br><br>\n <span style=\"font-size:1rem; color:rgba(240,208,96,0.7);\">Le cardinal de \u211d, not\u00e9 \ud835\udd20, est strictement sup\u00e9rieur \u00e0 \u2135\u2080.<\/span>\n <\/div>\n <\/div>\n\n <h2>La hi\u00e9rarchie des infinis<\/h2>\n\n <p>\n Et ce n’est pas tout. Cantor a montr\u00e9 qu’on peut construire des infinis de plus en plus grands, sans jamais s’arr\u00eater. Pour tout ensemble A, l’ensemble de ses parties (not\u00e9 \ud835\udcab(A)) a un cardinal strictement sup\u00e9rieur \u00e0 celui de A.\n <\/p>\n\n <div class=\"formule\">|\ud835\udcab(A)| > |A| \u2014 toujours, m\u00eame si A est infini<\/div>\n\n <p>\n Cela cr\u00e9e une hi\u00e9rarchie infinie d’infinis :\n <\/p>\n\n <div class=\"infinis-scale\">\n <div class=\"infini-row\">\n <div class=\"infini-label\" style=\"color:#3a1a5a; font-size:0.9rem;\">\u2135\u2080<\/div>\n <div class=\"infini-bar\" style=\"width:25%; background:#3a1a5a; font-size:0.8rem;\">\u2115, \u2124, \u211a<\/div>\n <div class=\"infini-desc\">Entiers, relatifs, fractions \u2014 tous pareils<\/div>\n <\/div>\n <div class=\"infini-row\">\n <div class=\"infini-label\" style=\"color:#5a1a9a; font-size:0.9rem;\">\ud835\udd20 = 2^\u2135\u2080<\/div>\n <div class=\"infini-bar\" style=\"width:50%; background:#5a1a9a; font-size:0.8rem;\">\u211d, intervalles, courbes<\/div>\n <div class=\"infini-desc\">Nombres r\u00e9els \u2014 strictement plus grand que \u2135\u2080<\/div>\n <\/div>\n <div class=\"infini-row\">\n <div class=\"infini-label\" style=\"color:#7a2aba; font-size:0.9rem;\">2^\ud835\udd20<\/div>\n <div class=\"infini-bar\" style=\"width:70%; background:#7a2aba; font-size:0.8rem;\">\ud835\udcab(\u211d) \u2014 parties de \u211d<\/div>\n <div class=\"infini-desc\">Encore plus grand \u2014 et on peut continuer…<\/div>\n <\/div>\n <div class=\"infini-row\">\n <div class=\"infini-label\" style=\"color:#9a4adf; font-size:0.9rem;\">2^(2^\ud835\udd20)<\/div>\n <div class=\"infini-bar\" style=\"width:90%; background:#9a4adf; font-size:0.8rem;\">\ud835\udcab(\ud835\udcab(\u211d))…<\/div>\n <div class=\"infini-desc\">Et encore plus… \u00e0 l’infini !<\/div>\n <\/div>\n <\/div>\n\n <h2>L’hypoth\u00e8se du continu \u2014 le myst\u00e8re non r\u00e9solu<\/h2>\n\n <p>\n Cantor a pos\u00e9 une question qui est rest\u00e9e ouverte pendant plus d’un si\u00e8cle : <strong>existe-t-il un infini entre \u2135\u2080 et \ud835\udd20 ?<\/strong> Autrement dit, y a-t-il un niveau d’infini entre celui des entiers et celui des r\u00e9els ?\n <\/p>\n\n <p>\n Cette question s’appelle l’<strong>hypoth\u00e8se du continu<\/strong>. La r\u00e9ponse est l’une des plus troublantes de toute la logique math\u00e9matique : <strong>on ne peut ni la prouver ni la r\u00e9futer<\/strong> dans les axiomes standard des math\u00e9matiques. C’est une question ind\u00e9cidable \u2014 elle d\u00e9passe les limites de notre syst\u00e8me formel.\n <\/p>\n\n <blockquote>\n L’infini est l\u00e0 o\u00f9 les math\u00e9matiques rencontrent la philosophie. Et Cantor a montr\u00e9 que m\u00eame l\u00e0, la logique peut nous guider \u2014 jusqu’\u00e0 une certaine fronti\u00e8re, au-del\u00e0 de laquelle le myst\u00e8re reste entier.\n <cite>\u2014 R\u00e9flexion sur les limites de la connaissance math\u00e9matique<\/cite>\n <\/blockquote>\n\n <h2>Le code Python \u2014 explorer les ensembles infinis<\/h2>\n\n <div class=\"code-block\">\n <span class=\"code-label\">Python<\/span>\n<span class=\"code-comment\"># Explorer les concepts de Cantor avec Python<\/span>\n<span class=\"code-keyword\">import<\/span> random\n<span class=\"code-keyword\">from<\/span> fractions <span class=\"code-keyword\">import<\/span> Fraction\n\n<span class=\"code-comment\"># — D\u00e9monstration 1 : Bijection N \u2192 Pairs —<\/span>\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0Bijection \u2115 \u2192 Nombres pairs (n \u2192 2n) :\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb{‘n’:>4} \u2192 {‘2n’:>6}\u00a0\u00bb<\/span>)\n<span class=\"code-keyword\">for<\/span> n <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">range<\/span>(<span class=\"code-number\">1<\/span>, <span class=\"code-number\">11<\/span>):\n <span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb{n:>4} \u2192 {2*n:>6}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># — D\u00e9monstration 2 : D\u00e9nombrement des rationnels —<\/span>\n<span class=\"code-comment\"># Parcours diagonal du tableau p\/q<\/span>\n<span class=\"code-keyword\">def<\/span> <span class=\"code-function\">rationnels_diagonale<\/span>(limite):\n <span class=\"code-string\">\u00ab\u00a0\u00a0\u00bb\u00a0\u00bbG\u00e9n\u00e8re les fractions positives par diagonale\u00a0\u00bb\u00a0\u00bb\u00a0\u00bb<\/span>\n rationnels = []\n vus = <span class=\"code-function\">set<\/span>()\n diagonale = <span class=\"code-number\">2<\/span>\n <span class=\"code-keyword\">while<\/span> <span class=\"code-function\">len<\/span>(rationnels) < limite:\n <span class=\"code-keyword\">for<\/span> p <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">range<\/span>(<span class=\"code-number\">1<\/span>, diagonale):\n q = diagonale – p\n f = Fraction(p, q)\n <span class=\"code-keyword\">if<\/span> f <span class=\"code-keyword\">not<\/span> <span class=\"code-keyword\">in<\/span> vus:\n vus.<span class=\"code-function\">add<\/span>(f)\n rationnels.<span class=\"code-function\">append<\/span>(f)\n <span class=\"code-keyword\">if<\/span> <span class=\"code-function\">len<\/span>(rationnels) >= limite:\n <span class=\"code-keyword\">break<\/span>\n diagonale += <span class=\"code-number\">1<\/span>\n <span class=\"code-keyword\">return<\/span> rationnels\n\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0\\nD\u00e9nombrement des rationnels par diagonale :\u00a0\u00bb<\/span>)\nrats = <span class=\"code-function\">rationnels_diagonale<\/span>(<span class=\"code-number\">20<\/span>)\n<span class=\"code-keyword\">for<\/span> i, r <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">enumerate<\/span>(rats, <span class=\"code-number\">1<\/span>):\n <span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb r_{i:02d} = {r}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># — D\u00e9monstration 3 : Argument diagonal de Cantor —<\/span>\n<span class=\"code-keyword\">def<\/span> <span class=\"code-function\">cantor_diagonal<\/span>(liste_reels, n_decimales=<span class=\"code-number\">10<\/span>):\n <span class=\"code-string\">\u00ab\u00a0\u00a0\u00bb\u00a0\u00bb\n Construit un r\u00e9el absent de la liste\n par l’argument diagonal de Cantor\n \u00ab\u00a0\u00a0\u00bb\u00a0\u00bb<\/span>\n nombre_diagonal = <span class=\"code-string\">\u00ab\u00a00.\u00a0\u00bb<\/span>\n <span class=\"code-keyword\">for<\/span> i, r <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">enumerate<\/span>(liste_reels[:n_decimales]):\n <span class=\"code-comment\"># Extrait le i-\u00e8me chiffre d\u00e9cimal de r_i<\/span>\n decimales = <span class=\"code-string\">f\u00a0\u00bb{r:.{n_decimales}f}\u00a0\u00bb<\/span>.<span class=\"code-function\">split<\/span>(<span class=\"code-string\">‘.’<\/span>)[<span class=\"code-number\">1<\/span>]\n chiffre = <span class=\"code-function\">int<\/span>(decimales[i]) <span class=\"code-keyword\">if<\/span> i < <span class=\"code-function\">len<\/span>(decimales) <span class=\"code-keyword\">else<\/span> <span class=\"code-number\">0<\/span>\n <span class=\"code-comment\"># R\u00e8gle : si chiffre = 4 \u2192 5, sinon \u2192 4<\/span>\n nouveau = <span class=\"code-string\">‘5’<\/span> <span class=\"code-keyword\">if<\/span> chiffre == <span class=\"code-number\">4<\/span> <span class=\"code-keyword\">else<\/span> <span class=\"code-string\">‘4’<\/span>\n nombre_diagonal += nouveau\n <span class=\"code-keyword\">return<\/span> nombre_diagonal\n\n<span class=\"code-comment\"># Liste \u00ab\u00a0suppos\u00e9e compl\u00e8te\u00a0\u00bb de r\u00e9els al\u00e9atoires<\/span>\nrandom.<span class=\"code-function\">seed<\/span>(<span class=\"code-number\">42<\/span>)\nliste = [random.<span class=\"code-function\">random<\/span>() <span class=\"code-keyword\">for<\/span> _ <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">range<\/span>(<span class=\"code-number\">8<\/span>)]\n\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0\\nArgument diagonal de Cantor :\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0Liste suppos\u00e9e compl\u00e8te :\u00a0\u00bb<\/span>)\n<span class=\"code-keyword\">for<\/span> i, r <span class=\"code-keyword\">in<\/span> <span class=\"code-function\">enumerate<\/span>(liste, <span class=\"code-number\">1<\/span>):\n <span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb r_{i} = {r:.10f}\u00a0\u00bb<\/span>)\n\nabsent = <span class=\"code-function\">cantor_diagonal<\/span>(liste)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb\\nNombre construit par diagonale : {absent}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0\u2192 Ce nombre est absent de la liste par construction !\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">\u00ab\u00a0\u2192 Donc aucune liste ne peut \u00eatre compl\u00e8te. CQFD.\u00a0\u00bb<\/span>)\n <\/div>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9 \u2014 du coll\u00e8ge au sup\u00e9rieur<\/h3>\n <p><strong>Niveau coll\u00e8ge :<\/strong> Construis une bijection entre les entiers naturels \u2115 = {0, 1, 2, 3…} et les entiers impairs {1, 3, 5, 7…}. Donne la r\u00e8gle explicite.<\/p>\n <p><strong>Niveau lyc\u00e9e :<\/strong> Montre que l’ensemble \u2124 des entiers relatifs (…-2, -1, 0, 1, 2…) est d\u00e9nombrable en construisant une bijection avec \u2115.<\/p>\n <p><strong>Niveau sup\u00e9rieur :<\/strong> En utilisant l’argument diagonal de Cantor, explique pourquoi l’ensemble de toutes les suites infinies de 0 et de 1 n’est pas d\u00e9nombrable.<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>Niveau coll\u00e8ge :<\/strong><\/p>\n <p>R\u00e8gle : n \u2192 2n + 1<\/p>\n <p>0 \u2192 1, 1 \u2192 3, 2 \u2192 5, 3 \u2192 7, 4 \u2192 9…<\/p>\n <p>C’est bien une bijection : chaque impair vient d’exactement un entier naturel. \u2705<\/p>\n\n <p><strong>Niveau lyc\u00e9e :<\/strong><\/p>\n <p>On liste \u2124 en \u00ab\u00a0zigzaguant\u00a0\u00bb autour de 0 :<\/p>\n <p>0 \u2192 0, 1 \u2192 1, 2 \u2192 -1, 3 \u2192 2, 4 \u2192 -2, 5 \u2192 3, 6 \u2192 -3…<\/p>\n <p>R\u00e8gle formelle : si n est pair \u2192 n\/2 ; si n est impair \u2192 -(n+1)\/2<\/p>\n <p>Chaque entier relatif appara\u00eet exactement une fois \u2192 bijection \u2192 \u2124 est d\u00e9nombrable \u2705<\/p>\n\n <p><strong>Niveau sup\u00e9rieur :<\/strong><\/p>\n <p>Supposons qu’on peut lister toutes les suites : s\u2081, s\u2082, s\u2083… o\u00f9 chaque s\u1d62 est une suite infinie de 0 et 1.<\/p>\n <p>On construit une nouvelle suite d en prenant le i-\u00e8me \u00e9l\u00e9ment de s\u1d62 et en le changeant (0\u21921, 1\u21920).<\/p>\n <p>d diff\u00e8re de s\u2081 en position 1, de s\u2082 en position 2, de s\u2083 en position 3… donc d diff\u00e8re de toute suite de la liste.<\/p>\n <p>d est une suite valide (de 0 et 1) qui n’est pas dans la liste \u2192 contradiction \u2192 l’ensemble n’est pas d\u00e9nombrable. \u2705<\/p>\n <\/div>\n <\/div>\n\n <div class=\"conclusion\">\n <h2>Ce qu’on retient<\/h2>\n <p>\n L’infini n’est pas une montagne unique \u2014 c’est une cha\u00eene de montagnes sans fin. \u2135\u2080, \ud835\udd20, 2^\ud835\udd20… chaque infini est strictement plus grand que le pr\u00e9c\u00e9dent, et il n’y a pas de sommet. Cantor a ouvert une porte que personne n’avait os\u00e9 franchir avant lui \u2014 et derri\u00e8re cette porte, les math\u00e9matiques sont devenues plus vastes, plus profondes, et plus \u00e9tranges que quiconque ne l’avait imagin\u00e9.\n <\/p>\n <p>\n La prochaine fois qu’un \u00e9l\u00e8ve te dit \u00ab\u00a0il y a une infinit\u00e9 de nombres\u00a0\u00bb, tu pourras lui r\u00e9pondre avec le sourire : \u00ab\u00a0Oui \u2014 mais de quelle taille d’infini parles-tu ?\u00a0\u00bb\n <\/p>\n <p>\n Dans le prochain article, nous plongerons dans <strong>les fractales<\/strong> \u2014 ces formes g\u00e9om\u00e9triques infiniment complexes qui se cachent dans les c\u00f4tes d’Afrique, les arbres, les poumons humains, et les march\u00e9s financiers.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>Pourquoi l’infini a plusieurs tailles \u2014 MathsVivantes Philosophie des maths \u00b7 Article #15 Pourquoi l’infinia plusieurs tailles \u2014et certains sont […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[8],"tags":[],"class_list":["post-136","post","type-post","status-publish","format-standard","hentry","category-fondements"],"yoast_head":"\n<title>La taille de l'Infini - AfrikIntelligent Pourquoi l'infini a plusieurs tailles - maths vivants<\/title>\n<meta name=\"description\" content=\"Tous les infinis ne sont pas \u00e9gaux . Cantor a prouv\u00e9 qu'il existe des infinis plus grands . D\u00e9couvrez l'argument diagonal et le vertige des math\u00e9matiques .\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/vmi3217629.contaboserver.net\/?p=136\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"La taille de l'Infini - AfrikIntelligent Pourquoi l'infini a plusieurs tailles - maths vivants\" \/>\n<meta property=\"og:description\" content=\"Tous les infinis ne sont pas \u00e9gaux . Cantor a prouv\u00e9 qu'il existe des infinis plus grands . D\u00e9couvrez l'argument diagonal et le vertige des math\u00e9matiques .\" \/>\n<meta property=\"og:url\" content=\"https:\/\/vmi3217629.contaboserver.net\/?p=136\" \/>\n<meta property=\"og:site_name\" content=\"AfrikIntelligent\" \/>\n<meta property=\"article:published_time\" content=\"2026-04-29T06:15:48+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-04-29T06:15:49+00:00\" \/>\n<meta name=\"author\" content=\"Odilon\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"Odilon\" \/>\n\t<meta name=\"twitter:label2\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data2\" content=\"11 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136\"},\"author\":{\"name\":\"Odilon\",\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/#\\\/schema\\\/person\\\/a3da39574d729b6b3ff7514ccc055923\"},\"headline\":\"La taille de l’Infini\",\"datePublished\":\"2026-04-29T06:15:48+00:00\",\"dateModified\":\"2026-04-29T06:15:49+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136\"},\"wordCount\":2062,\"commentCount\":0,\"articleSection\":[\"Fondements\"],\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136\",\"url\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136\",\"name\":\"La taille de l'Infini - AfrikIntelligent Pourquoi l'infini a plusieurs tailles - maths vivants\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/#website\"},\"datePublished\":\"2026-04-29T06:15:48+00:00\",\"dateModified\":\"2026-04-29T06:15:49+00:00\",\"author\":{\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/#\\\/schema\\\/person\\\/a3da39574d729b6b3ff7514ccc055923\"},\"description\":\"Tous les infinis ne sont pas \u00e9gaux . Cantor a prouv\u00e9 qu'il existe des infinis plus grands . D\u00e9couvrez l'argument diagonal et le vertige des math\u00e9matiques .\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136#breadcrumb\"},\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?p=136#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Accueil\",\"item\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"La taille de l’Infini\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/#website\",\"url\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/\",\"name\":\"AfrikIntelligent\",\"description\":\"Append autrement\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"fr-FR\"},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/#\\\/schema\\\/person\\\/a3da39574d729b6b3ff7514ccc055923\",\"name\":\"Odilon\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/57bcd6a76fa395ae5e994943b66ccd3dd7e71cd0b1b77b351986e88c27801523?s=96&d=mm&r=g\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/57bcd6a76fa395ae5e994943b66ccd3dd7e71cd0b1b77b351986e88c27801523?s=96&d=mm&r=g\",\"contentUrl\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/57bcd6a76fa395ae5e994943b66ccd3dd7e71cd0b1b77b351986e88c27801523?s=96&d=mm&r=g\",\"caption\":\"Odilon\"},\"sameAs\":[\"https:\\\/\\\/vmi3217629.contaboserver.net\"],\"url\":\"https:\\\/\\\/vmi3217629.contaboserver.net\\\/?author=1\"}]}<\/script>\n","yoast_head_json":{"title":"La taille de l'Infini - AfrikIntelligent Pourquoi l'infini a plusieurs tailles - maths vivants","description":"Tous les infinis ne sont pas \u00e9gaux . Cantor a prouv\u00e9 qu'il existe des infinis plus grands . D\u00e9couvrez l'argument diagonal et le vertige des math\u00e9matiques .","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/vmi3217629.contaboserver.net\/?p=136","og_locale":"fr_FR","og_type":"article","og_title":"La taille de l'Infini - AfrikIntelligent Pourquoi l'infini a plusieurs tailles - maths vivants","og_description":"Tous les infinis ne sont pas \u00e9gaux . Cantor a prouv\u00e9 qu'il existe des infinis plus grands . D\u00e9couvrez l'argument diagonal et le vertige des math\u00e9matiques .","og_url":"https:\/\/vmi3217629.contaboserver.net\/?p=136","og_site_name":"AfrikIntelligent","article_published_time":"2026-04-29T06:15:48+00:00","article_modified_time":"2026-04-29T06:15:49+00:00","author":"Odilon","twitter_card":"summary_large_image","twitter_misc":{"\u00c9crit par":"Odilon","Dur\u00e9e de lecture estim\u00e9e":"11 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/vmi3217629.contaboserver.net\/?p=136#article","isPartOf":{"@id":"https:\/\/vmi3217629.contaboserver.net\/?p=136"},"author":{"name":"Odilon","@id":"https:\/\/vmi3217629.contaboserver.net\/#\/schema\/person\/a3da39574d729b6b3ff7514ccc055923"},"headline":"La taille de l’Infini","datePublished":"2026-04-29T06:15:48+00:00","dateModified":"2026-04-29T06:15:49+00:00","mainEntityOfPage":{"@id":"https:\/\/vmi3217629.contaboserver.net\/?p=136"},"wordCount":2062,"commentCount":0,"articleSection":["Fondements"],"inLanguage":"fr-FR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/vmi3217629.contaboserver.net\/?p=136#respond"]}]},{"@type":"WebPage","@id":"https:\/\/vmi3217629.contaboserver.net\/?p=136","url":"https:\/\/vmi3217629.contaboserver.net\/?p=136","name":"La taille de l'Infini - AfrikIntelligent Pourquoi l'infini a plusieurs tailles - maths vivants","isPartOf":{"@id":"https:\/\/vmi3217629.contaboserver.net\/#website"},"datePublished":"2026-04-29T06:15:48+00:00","dateModified":"2026-04-29T06:15:49+00:00","author":{"@id":"https:\/\/vmi3217629.contaboserver.net\/#\/schema\/person\/a3da39574d729b6b3ff7514ccc055923"},"description":"Tous les infinis ne sont pas \u00e9gaux . Cantor a prouv\u00e9 qu'il existe des infinis plus grands . D\u00e9couvrez l'argument diagonal et le vertige des math\u00e9matiques .","breadcrumb":{"@id":"https:\/\/vmi3217629.contaboserver.net\/?p=136#breadcrumb"},"inLanguage":"fr-FR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/vmi3217629.contaboserver.net\/?p=136"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/vmi3217629.contaboserver.net\/?p=136#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Accueil","item":"https:\/\/vmi3217629.contaboserver.net\/"},{"@type":"ListItem","position":2,"name":"La taille de l’Infini"}]},{"@type":"WebSite","@id":"https:\/\/vmi3217629.contaboserver.net\/#website","url":"https:\/\/vmi3217629.contaboserver.net\/","name":"AfrikIntelligent","description":"Append autrement","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/vmi3217629.contaboserver.net\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"fr-FR"},{"@type":"Person","@id":"https:\/\/vmi3217629.contaboserver.net\/#\/schema\/person\/a3da39574d729b6b3ff7514ccc055923","name":"Odilon","image":{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/secure.gravatar.com\/avatar\/57bcd6a76fa395ae5e994943b66ccd3dd7e71cd0b1b77b351986e88c27801523?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/57bcd6a76fa395ae5e994943b66ccd3dd7e71cd0b1b77b351986e88c27801523?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/57bcd6a76fa395ae5e994943b66ccd3dd7e71cd0b1b77b351986e88c27801523?s=96&d=mm&r=g","caption":"Odilon"},"sameAs":["https:\/\/vmi3217629.contaboserver.net"],"url":"https:\/\/vmi3217629.contaboserver.net\/?author=1"}]}},"uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false},"uagb_author_info":{"display_name":"Odilon","author_link":"https:\/\/vmi3217629.contaboserver.net\/?author=1"},"uagb_comment_info":0,"uagb_excerpt":"Pourquoi l’infini a plusieurs tailles \u2014 MathsVivantes Philosophie des maths \u00b7 Article #15 Pourquoi l’infinia plusieurs tailles \u2014et certains sont […]","_links":{"self":[{"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/posts\/136","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=136"}],"version-history":[{"count":1,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/posts\/136\/revisions"}],"predecessor-version":[{"id":137,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/posts\/136\/revisions\/137"}],"wp:attachment":[{"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=136"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=136"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=136"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}