Timeline for All sets of rational numbers are bigger than the set containing infinite integers - or are they?
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| Jun 12, 2020 at 10:38 | history | edited | CommunityBot |
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| Apr 13, 2017 at 12:21 | history | edited | CommunityBot |
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| S Apr 14, 2015 at 22:39 | history | suggested | Electric Coffee | CC BY-SA 3.0 |
it's not a theory, it's a hypothesis, and unless he means professor, it's proof
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| Apr 14, 2015 at 21:56 | review | Suggested edits | |||
| S Apr 14, 2015 at 22:39 | |||||
| Apr 14, 2015 at 18:10 | answer | added | David K | timeline score: 0 | |
| Apr 14, 2015 at 18:01 | comment | added | David K | I think the term you are looking for (to describe different "types" of infinity) is cardinality. You set out to prove that the non-negative integers are of the exact same cardinality as the set of rationals in the interval $(0,1].$ | |
| Apr 14, 2015 at 17:37 | answer | added | Michael Hardy | timeline score: 2 | |
| Apr 14, 2015 at 17:37 | comment | added | Winther | It is possible to construct a bijection between all rationals in $[0,1)$ and all rationals in any interval $[a,b)$ with $b>a$ (also if $a=-\infty$, $b=\infty$). This does not follows directly from this proof though. | |
| Apr 14, 2015 at 17:28 | comment | added | user3265569 | Thanks Winther. One last thing: Does this mean, since can map any set of rational numbers, that there too are exactly as many rational numbers in the set [0;1[ as in the set of rational numbers [0;10[ and [0;∞] | |
| Apr 14, 2015 at 17:20 | answer | added | NovaDenizen | timeline score: 0 | |
| Apr 14, 2015 at 17:19 | comment | added | Winther | Note that $0.1 = 1/10$, $0.2 = 1/5$ etc. so I would remove the first part (it is covered by the latter so you don't need it - it just complicates stuff). But that method should work (small typo: $4/6 = 2/3$ has been listen before). You are guaranteed to go through all rational numbers in $[0,1)$ and map them one-to-one to the positive integers giving you a bijection. | |
| Apr 14, 2015 at 17:05 | comment | added | user3265569 | Thanks. I've made an edit. Care to take a look? | |
| Apr 14, 2015 at 17:05 | history | edited | user3265569 | CC BY-SA 3.0 |
added 1162 characters in body
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| Apr 14, 2015 at 16:37 | comment | added | Winther | You have to define what that means. Is it $\ldots 824175824175824175$, what number is this? It's not a real number. As a sidenote one can make sense of number of that form (see p-adic numbers but that has nothing to do with the construction you are trying to make. | |
| Apr 14, 2015 at 16:31 | comment | added | user3265569 | Will this: $\overline{824175}824175$ make more sense then? | |
| Apr 14, 2015 at 16:26 | comment | added | Winther | What is $824175\overline{824175}$ supposed to mean? Infinite repetion to the right ($824175824175824175\ldots$) is not a real number. The reason $0.571428\overline{571428}$ makes sense is that it is represented by a series that converges. | |
| Apr 14, 2015 at 16:23 | history | edited | user3265569 | CC BY-SA 3.0 |
added 6 characters in body
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| Apr 14, 2015 at 16:03 | history | edited | kennytm | CC BY-SA 3.0 |
overlines.
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| Apr 14, 2015 at 15:55 | review | First posts | |||
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| Apr 14, 2015 at 15:55 | history | asked | user3265569 | CC BY-SA 3.0 |