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real_numbers [2016/01/19 22:18] nikolaj |
real_numbers [2016/01/19 22:18] nikolaj |
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| == Specker sequence == | == Specker sequence == | ||
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| - | >>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 $i$. | ||
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| 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 $) |
| $a_n = \sum_{i+s=n} \frac {1} {2^i} h(i,s)$ | $a_n = \sum_{i+s=n} \frac {1} {2^i} h(i,s)$ | ||
