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 diagonal_construction [2016/04/05 00:43]nikolaj diagonal_construction [2019/08/25 13:48] (current)nikolaj fix $Both sides previous revision Previous revision 2019/08/25 13:48 nikolaj fix$2016/04/05 00:43 nikolaj 2016/04/05 00:21 nikolaj 2016/01/13 10:28 nikolaj 2016/01/11 10:52 nikolaj 2016/01/11 10:51 nikolaj 2016/01/11 10:51 nikolaj 2016/01/11 10:47 nikolaj 2016/01/11 10:46 nikolaj 2016/01/11 10:41 nikolaj 2016/01/11 10:40 nikolaj 2016/01/11 10:40 nikolaj 2016/01/11 10:39 nikolaj 2016/01/11 10:37 nikolaj 2016/01/11 10:32 nikolaj 2015/12/28 13:19 nikolaj 2015/12/28 13:18 nikolaj 2014/08/22 16:39 nikolaj 2014/08/22 16:37 nikolaj 2014/08/22 16:34 nikolaj 2014/08/22 16:34 nikolaj 2014/08/22 16:33 nikolaj 2014/08/22 16:31 nikolaj 2014/08/22 16:30 nikolaj 2014/08/22 16:24 nikolaj 2014/08/22 16:24 nikolaj 2014/08/22 16:23 nikolaj 2019/08/25 13:48 nikolaj fix $2016/04/05 00:43 nikolaj 2016/04/05 00:21 nikolaj 2016/01/13 10:28 nikolaj 2016/01/11 10:52 nikolaj 2016/01/11 10:51 nikolaj 2016/01/11 10:51 nikolaj 2016/01/11 10:47 nikolaj 2016/01/11 10:46 nikolaj 2016/01/11 10:41 nikolaj 2016/01/11 10:40 nikolaj 2016/01/11 10:40 nikolaj 2016/01/11 10:39 nikolaj 2016/01/11 10:37 nikolaj 2016/01/11 10:32 nikolaj 2015/12/28 13:19 nikolaj 2015/12/28 13:18 nikolaj 2014/08/22 16:39 nikolaj 2014/08/22 16:37 nikolaj 2014/08/22 16:34 nikolaj 2014/08/22 16:34 nikolaj 2014/08/22 16:33 nikolaj 2014/08/22 16:31 nikolaj 2014/08/22 16:30 nikolaj 2014/08/22 16:24 nikolaj 2014/08/22 16:24 nikolaj 2014/08/22 16:23 nikolaj 2014/08/22 16:22 nikolaj 2014/08/22 16:22 nikolaj 2014/08/22 16:21 nikolaj 2014/03/21 11:11 external edit2013/08/23 21:54 nikolaj 2013/06/20 16:01 nikolaj 2013/06/20 15:56 nikolaj 2013/06/20 15:56 nikolaj 2013/06/20 15:54 nikolaj 2013/06/20 15:50 nikolaj 2013/06/20 15:49 nikolaj 2013/06/20 15:44 nikolaj 2013/06/20 01:46 nikolaj 2013/06/20 01:44 nikolaj 2013/06/20 01:36 nikolaj 2013/06/20 01:35 nikolaj 2013/06/18 22:59 nikolaj old revision restored (2013/06/18 22:52)2013/06/18 22:59 nikolaj 2013/06/18 22:52 nikolaj 2013/06/18 22:50 nikolaj 2013/06/18 22:49 nikolaj 2013/06/18 22:48 nikolaj 2013/06/18 22:47 nikolaj created Line 32: Line 32: This holds always. This holds always. - Now, specifically,​ for any$x\in D_f$and using$X=D_f$and$Y=f(x)$, the left hand side reads$x\in D_f\land x\notin f(x)$(which by definition of$D_f$is the same as just$x\in D_f$) and the right hand side reads$\neg(f(x)=D_f)$. The same happens for$x\in C$but not in$D_f and where we then switch $X$ and $Y$. This means $\nexists x\ (f(x)=D_f)$. + Now, specifically,​ for any $x\in D_f$ and using $X=D_f$ and $Y=f(x)$, the left hand side reads $x\in D_f\land x\notin f(x)$ (which by definition of $D_f$ is the same as just $x\in D_f$) and the right hand side reads $\neg(f(x)=D_f)$. The same happens for $x\in C$ but not in $D_f$ and where we then switch $X$ and $Y$. This means $\nexists x\ (f(x)=D_f)$. - Since $D_f\subseteq C$, i.e. $D_f \in \mathcal P(C) =\text{codom}(f)$,​ we see that no such $f$ is a surjection, let alone a bijection. So the cardinality of any set is less than that of its power set. $\Box$ + Since $D_f\subseteq C$, i.e. $D_f \in \mathcal P(C) = \mathrm{codom} (f)$, we see that no such $f$ is a surjection, let alone a bijection. So the cardinality of any set is less than that of its power set. $\Box$ If, in a similar spirit, $C,f$ are taken to be a sequence and an enumeration of sequences, then we see that there is at least once sequence which escapes enumeration. This further translates to the uncountability of real numbers. ​ If, in a similar spirit, $C,f$ are taken to be a sequence and an enumeration of sequences, then we see that there is at least once sequence which escapes enumeration. This further translates to the uncountability of real numbers. ​