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 diff --git a/notes-mitro206.tex b/notes-mitro206.texindex 2c4cf75..ba84960 100644--- a/notes-mitro206.tex+++ b/notes-mitro206.tex@@ -3165,6 +3165,140 @@ $G/\equiv$, on a bien $f(x) = f(x')$ ssi $x\equiv x'$). % % +\section{Introduction aux ordinaux}++\subsection{Explication intuitive}++\thingy Les ordinaux sont une sorte de nombres, totalement ordonnés et+même « bien-ordonnés », qui généralisent les entiers naturels en+allant « au-delà de l'infini » : les entiers naturels+$0,1,2,3,4,\ldots$ sont en particulier des ordinaux (ce sont les plus+petits), mais il existe un ordinal qui vient après eux, à+savoir $\omega$, qui est lui-même suivi de+$\omega+1,\omega+2,\omega+3,\ldots$, après quoi vient $\omega\cdot 2$+(ou simplement $\omega 2$), et beaucoup d'autres choses.++\begin{center}+\begin{tikzpicture}+\begin{scope}[line width=1.5pt,cap=round,join=round]+% x = 10*(1-0.8^k*(1-0.2*(1-0.8^n)))+% y = 2*0.8^k*0.8^n+\draw (0.00000,2.00000) -- (0.00000,-2.00000);+\draw (0.40000,1.60000) -- (0.40000,-1.60000);+\draw (0.72000,1.28000) -- (0.72000,-1.28000);+\draw (0.97600,1.02400) -- (0.97600,-1.02400);+\draw (1.18080,0.81920) -- (1.18080,-0.81920);+\draw (1.34464,0.65536) -- (1.34464,-0.65536);+\draw (1.47571,0.52429) -- (1.47571,-0.52429);+\draw (1.58057,0.41943) -- (1.58057,-0.41943);+\draw (1.66446,0.33554) -- (1.66446,-0.33554);+\draw (1.73156,0.26844) -- (1.73156,-0.26844);+\draw[fill] (1.78525,0.21475) -- (1.78525,-0.21475) -- (2.00000,0.00000) -- (1.78525,0.21475);+\draw (2.00000,1.60000) -- (2.00000,-1.60000);+\draw (2.32000,1.28000) -- (2.32000,-1.28000);+\draw (2.57600,1.02400) -- (2.57600,-1.02400);+\draw (2.78080,0.81920) -- (2.78080,-0.81920);+\draw (2.94464,0.65536) -- (2.94464,-0.65536);+\draw (3.07571,0.52429) -- (3.07571,-0.52429);+\draw (3.18057,0.41943) -- (3.18057,-0.41943);+\draw (3.26446,0.33554) -- (3.26446,-0.33554);+\draw (3.33156,0.26844) -- (3.33156,-0.26844);+\draw (3.38525,0.21475) -- (3.38525,-0.21475);+\draw[fill] (3.42820,0.17180) -- (3.42820,-0.17180) -- (3.60000,0.00000) -- (3.42820,0.17180);+\draw (3.60000,1.28000) -- (3.60000,-1.28000);+\draw (3.85600,1.02400) -- (3.85600,-1.02400);+\draw (4.06080,0.81920) -- (4.06080,-0.81920);+\draw (4.22464,0.65536) -- (4.22464,-0.65536);+\draw (4.35571,0.52429) -- (4.35571,-0.52429);+\draw (4.46057,0.41943) -- (4.46057,-0.41943);+\draw (4.54446,0.33554) -- (4.54446,-0.33554);+\draw (4.61156,0.26844) -- (4.61156,-0.26844);+\draw (4.66525,0.21475) -- (4.66525,-0.21475);+\draw (4.70820,0.17180) -- (4.70820,-0.17180);+\draw[fill] (4.74256,0.13744) -- (4.74256,-0.13744) -- (4.88000,0.00000) -- (4.74256,0.13744);+\draw (4.88000,1.02400) -- (4.88000,-1.02400);+\draw (5.08480,0.81920) -- (5.08480,-0.81920);+\draw (5.24864,0.65536) -- (5.24864,-0.65536);+\draw (5.37971,0.52429) -- (5.37971,-0.52429);+\draw (5.48457,0.41943) -- (5.48457,-0.41943);+\draw (5.56846,0.33554) -- (5.56846,-0.33554);+\draw (5.63556,0.26844) -- (5.63556,-0.26844);+\draw (5.68925,0.21475) -- (5.68925,-0.21475);+\draw (5.73220,0.17180) -- (5.73220,-0.17180);+\draw (5.76656,0.13744) -- (5.76656,-0.13744);+\draw[fill] (5.79405,0.10995) -- (5.79405,-0.10995) -- (5.90400,0.00000) -- (5.79405,0.10995);+\draw (5.90400,0.81920) -- (5.90400,-0.81920);+\draw (6.06784,0.65536) -- (6.06784,-0.65536);+\draw (6.19891,0.52429) -- (6.19891,-0.52429);+\draw (6.30377,0.41943) -- (6.30377,-0.41943);+\draw (6.38766,0.33554) -- (6.38766,-0.33554);+\draw (6.45476,0.26844) -- (6.45476,-0.26844);+\draw (6.50845,0.21475) -- (6.50845,-0.21475);+\draw (6.55140,0.17180) -- (6.55140,-0.17180);+\draw (6.58576,0.13744) -- (6.58576,-0.13744);+\draw (6.61325,0.10995) -- (6.61325,-0.10995);+\draw[fill] (6.63524,0.08796) -- (6.63524,-0.08796) -- (6.72320,0.00000) -- (6.63524,0.08796);+\draw (6.72320,0.65536) -- (6.72320,-0.65536);+\draw (6.85427,0.52429) -- (6.85427,-0.52429);+\draw (6.95913,0.41943) -- (6.95913,-0.41943);+\draw (7.04302,0.33554) -- (7.04302,-0.33554);+\draw (7.11012,0.26844) -- (7.11012,-0.26844);+\draw[fill] (7.16381,0.21475) -- (7.16381,-0.21475) -- (7.37856,0.00000) -- (7.16381,0.21475);+\draw (7.37856,0.52429) -- (7.37856,-0.52429);+\draw (7.48342,0.41943) -- (7.48342,-0.41943);+\draw (7.56730,0.33554) -- (7.56730,-0.33554);+\draw (7.63441,0.26844) -- (7.63441,-0.26844);+\draw (7.68810,0.21475) -- (7.68810,-0.21475);+\draw[fill] (7.73105,0.17180) -- (7.73105,-0.17180) -- (7.90285,0.00000) -- (7.73105,0.17180);+\draw (7.90285,0.41943) -- (7.90285,-0.41943);+\draw (7.98673,0.33554) -- (7.98673,-0.33554);+\draw (8.05384,0.26844) -- (8.05384,-0.26844);+\draw (8.10753,0.21475) -- (8.10753,-0.21475);+\draw (8.15048,0.17180) -- (8.15048,-0.17180);+\draw[fill] (8.18484,0.13744) -- (8.18484,-0.13744) -- (8.32228,0.00000) -- (8.18484,0.13744);+\draw (8.32228,0.33554) -- (8.32228,-0.33554);+\draw (8.38939,0.26844) -- (8.38939,-0.26844);+\draw (8.44307,0.21475) -- (8.44307,-0.21475);+\draw[fill] (8.48602,0.17180) -- (8.48602,-0.17180) -- (8.65782,0.00000) -- (8.48602,0.17180);+\draw (8.65782,0.26844) -- (8.65782,-0.26844);+\draw (8.71151,0.21475) -- (8.71151,-0.21475);+\draw (8.75446,0.17180) -- (8.75446,-0.17180);+\draw[fill] (8.78882,0.13744) -- (8.78882,-0.13744) -- (8.92626,0.00000) -- (8.78882,0.13744);+\draw (8.92626,0.21475) -- (8.92626,-0.21475);+\draw (8.96921,0.17180) -- (8.96921,-0.17180);+\draw (9.00357,0.13744) -- (9.00357,-0.13744);+\draw[fill] (9.03106,0.10995) -- (9.03106,-0.10995) -- (9.14101,0.00000) -- (9.03106,0.10995);+\draw (9.14101,0.17180) -- (9.14101,-0.17180);+\draw (9.17537,0.13744) -- (9.17537,-0.13744);+\draw[fill] (9.20285,0.10995) -- (9.22484,-0.08796) -- (9.31281,0.00000) -- (9.20285,0.10995);+\draw[fill] (9.31281,0.13744) -- (9.31281,-0.13744) -- (9.45024,0.00000) -- (9.31281,0.13744);+\draw[fill] (9.45024,0.10995) -- (9.45024,-0.10995) -- (9.56020,0.00000) -- (9.45024,0.10995);+\draw[fill] (9.56020,0.08796) -- (9.56020,-0.08796) -- (9.64816,0.00000) -- (9.56020,0.08796);+\draw[fill] (9.64816,0.07037) -- (9.64816,-0.07037) -- (9.71853,0.00000) -- (9.64816,0.07037);+\draw[fill] (9.71853,0.05629) -- (9.71853,-0.05629) -- (9.77482,0.00000) -- (9.71853,0.05629);+\draw[fill] (9.77482,0.04504) -- (9.77482,-0.04504) -- (9.81986,0.00000) -- (9.77482,0.04504);+\draw[fill] (9.81986,0.03603) -- (9.81986,-0.03603) -- (9.85588,0.00000) -- (9.81986,0.03603);+\draw[fill] (9.85588,0.02882) -- (9.85588,-0.02882) -- (9.88471,0.00000) -- (9.85588,0.02882);+\draw[fill] (9.88471,0.02306) -- (9.88471,-0.02306) -- (10.00000,0.00000) -- (9.88471,0.02306);+\end{scope}+\node[anchor=north] at (0.00000,-2.00000) {$0$};+\node[anchor=north] at (0.40000,-1.60000) {$1$};+\node[anchor=north] at (0.72000,-1.28000) {$2$};+\node[anchor=north] at (0.97600,-1.02400) {$3$};+\node[anchor=north] at (2.00000,-1.60000) {$\omega$};+\node[anchor=north] at (2.32000,-1.28000) {$\scriptscriptstyle \omega+1$};+\node[anchor=north] at (3.60000,-1.28000) {$\omega2$};+\node[anchor=north] at (4.88000,-1.02400) {$\omega3$};+\end{tikzpicture}+\\{\footnotesize (Une rangée de $\omega^2$ allumettes.)}+\end{center}++\thingy On pourra ajouter les ordinaux, et les multiplier, et même+élever un ordinal à la puissance d'un autre, mais il n'y aura pas de+soustraction ($\omega-1$ n'a pas de sens, en tout cas pas en tant+qu'ordinal, parce que $\omega$ est le plus petit ordinal infini).++ % % %