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real_numbers [2016/01/19 22:18] nikolaj |
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| - | /* graph */ | ||
| ===== Real numbers ===== | ===== Real numbers ===== | ||
| - | ==== Definition ==== | + | ==== Framework ==== |
| - | $ \mathbb{R} $ | + | $\dots,\,-\frac{\pi}{2},\,0,\,1,\dots$ |
| - | ==== Ramifications ==== | + | |
| - | ==== Context ==== | + | ${\mathbb R}$ |
| - | 13050403 | + | |
| - | == Requirements == | + | ----- |
| - | [[Natural numbers]] | + | |
| - | == Generalizations == | + | === Discussion === |
| - | == Computation == | + | >axiomatics and cardinality |
| - | ==== Elaboration ==== | + | |
| - | == Reference == | + | == Specker sequence == |
| - | [[http://en.wikipedia.org/wiki/Real_number|Real number on wikipedia]] | + | 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} \dfrac {h(i,s)} {2^i} $ | ||
| + | |||
| + | 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. | ||
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| + | And most real numbers are worse, really, because this number is at least definable. Most reals aren't even that. | ||
| + | |||
| + | === References === | ||
| + | Wikipedia: | ||
| + | [[http://en.wikipedia.org/wiki/Real_number|Real number]], | ||
| + | [[http://en.wikipedia.org/wiki/Construction_of_the_real_numbers|Construction of the real numbers]] | ||
| + | [[http://en.wikipedia.org/wiki/Computable_number|Computable number]] | ||
| + | [[http://en.wikipedia.org/wiki/Definable_real_number|Definable real number]] | ||
| + | |||
| + | ----- | ||
| + | === Related === | ||
| + | [[Logic]] | ||
