# Differences

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real_numbers [2014/12/27 19:32] nikolaj |
real_numbers [2016/01/19 22:18] nikolaj |
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=== Discussion === | === Discussion === | ||

>axiomatics and cardinality | >axiomatics and cardinality | ||

+ | |||

+ | == Specker sequence == | ||

+ | 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. | ||

+ | 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 $) | ||

+ | |||

+ | $a_n = \sum_{i+s=n} \frac {1} {2^i} h(i,s)$ | ||

+ | |||

+ | This is some number between 0 and 1. | ||

+ | |||

+ | But computing limit n to infinity (and thus s to infinity for all i) requires knowledge whether Turing machines ever halt, which we know to be impossible since the 30's. Thus this real number is not computable. | ||

+ | |||

+ | And most real numbers are worse, really, because this number is at least definable. Most reals aren't even that. | ||

=== References === | === References === |