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 real_numbers [2016/01/19 22:18]nikolaj real_numbers [2016/01/19 22:18]nikolaj Both sides previous revision Previous revision 2016/01/19 22:18 nikolaj 2016/01/19 22:18 nikolaj 2016/01/19 22:18 nikolaj 2014/12/27 19:32 nikolaj 2014/12/27 19:30 nikolaj old revision restored (2014/12/27 19:12)2013/05/04 23:43 nikolaj removed2013/05/04 23:15 nikolaj created 2016/01/19 22:18 nikolaj 2016/01/19 22:18 nikolaj 2016/01/19 22:18 nikolaj 2014/12/27 19:32 nikolaj 2014/12/27 19:30 nikolaj old revision restored (2014/12/27 19:12)2013/05/04 23:43 nikolaj removed2013/05/04 23:15 nikolaj created Last revision Both sides next revision Line 11: Line 11: == Specker sequence == == Specker sequence == - - >>​7793253 Choose a programming language and index all Turing machines (or executable program) by $i$. Choose a programming language and index all Turing machines (or executable program) by $i$. + Let $h(i,s)$ be $0$ or $1$, depending on whether the machine with index $i$ halts before $s$ steps. ​ Let $h(i,s)$ be $0$ or $1$, depending on whether the machine with index $i$ halts before $s$ steps. ​ For any given pair of numbers, you can indeed compute $h(i,s)$ by just running the program and wait $s$ time steps. For any given pair of numbers, you can indeed compute $h(i,s)$ by just running the program and wait $s$ time steps. - Define a sequence of rationals by having the n'th number given by the following sum (where you run through i and run it to step $s=n-i$) + Define a sequence of rationals by having the $n$'th number given by the following sum (where you run through i and run it to step $s=n-i$) $a_n = \sum_{i+s=n} \frac {1} {2^i} h(i,s)$ $a_n = \sum_{i+s=n} \frac {1} {2^i} h(i,s)$