Timeline for Counting Real Numbers
Current License: CC BY-SA 3.0
29 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 17, 2014 at 9:56 | comment | added | MGA | I'm not sure if you've come across this article, but it quite a long way for me when I was trying to understand un/countability: en.wikipedia.org/wiki/Cantor's_diagonal_argument | |
| Aug 13, 2013 at 14:46 | answer | added | Afrenett | timeline score: -2 | |
| Aug 13, 2013 at 14:33 | history | edited | Hans Lundmark | edited tags | |
| Aug 2, 2013 at 16:26 | comment | added | Haitham Gad | @HenningMakholm It's only hypothetically useful, just like the infinitely accurate $\pi$ or $\sqrt{2}$. | |
| Aug 2, 2013 at 12:34 | comment | added | hmakholm left over Monica | @HaithamGad: For the moment, until you define something you can do with it, it's just a string of digits. | |
| Aug 2, 2013 at 11:35 | comment | added | Haitham Gad | BTW: If 3333.... is not a natural number, what's the right name for it? (is it transfinite?) | |
| Aug 2, 2013 at 11:33 | comment | added | OR. | @GitGud Because the argument was supposed to define a surjection of the naturals. In such a table one would have to use several times the naturals but then it is missing the argument about how many times the naturals were used. | |
| Aug 2, 2013 at 11:21 | comment | added | Haitham Gad | @EuYu Got it. Thanks! (BTW, you can post this as an answer and I'll accept it). | |
| Aug 2, 2013 at 11:18 | comment | added | EuYu | @HaithamGad Yes, that's essentially the problem. The original table used by Cantor had an infinite number of columns (and rows), but the index of each column (and row) was finite. Right now you are considering a table in which the indices themselves need to be infinite. That's a whole different object altogether. | |
| Aug 2, 2013 at 11:17 | comment | added | Haitham Gad | @EuYu Ok, now it's starting to make sense. So the problem boils down to not being able to "count" a number with infinite digits, right? | |
| Aug 2, 2013 at 11:10 | comment | added | EuYu | @HaithamGad $333333\cdots$ is not a natural number. The set of natural numbers (i.e. $\mathbb{N}$) is infinite, but each given member of the set is finite. | |
| Aug 2, 2013 at 11:09 | comment | added | Haitham Gad | @EuYu Why? Isn't 33333... a natural number? | |
| Aug 2, 2013 at 11:08 | comment | added | Git Gud | @EuYu But now you're just talking too rigorously about an informal concept, namely infinite table. The OP is trying to convey an idea non-rigorously (and I tried to help him with that), but you're talking about it on a different level of formalism. The way I see it there are two ways to go about this: the OP formalizes his infinite table and we go from there or we keep things simple and informal and we can't use what you just said. | |
| Aug 2, 2013 at 11:05 | comment | added | Haitham Gad | @HagenvonEitzen n.m?! | |
| Aug 2, 2013 at 11:05 | comment | added | EuYu | @GitGud The purpose of the diagonal argument is to demonstrate that no such bijection exists. Yes, you can consider the "infinite'th" column, but then you no longer have a map involving $\mathbb{N}\times \mathbb{N}$. | |
| Aug 2, 2013 at 11:05 | comment | added | Hagen von Eitzen | Your table consists mostly of dots, so that's not even a foraml definition of a table. Can you specify explicitly which real number occurs in row $n$, column $m$? | |
| Aug 2, 2013 at 11:04 | comment | added | rurouniwallace | There are probably much better ways to answer this, but basically, you would "run out" of natural numbers before you manage to finish. | |
| Aug 2, 2013 at 11:04 | comment | added | Git Gud | @EuYu I suppose the OPs purpose is to find a bijection between $\Bbb R$ and $\Bbb N\times \Bbb N$. | |
| Aug 2, 2013 at 11:02 | review | First posts | |||
| Aug 2, 2013 at 11:02 | |||||
| Aug 2, 2013 at 11:01 | comment | added | Haitham Gad | @EuYu It should show up in the zeroth row and the infinite'th column :) just as the 333333.../100000.... rational number. | |
| Aug 2, 2013 at 11:01 | comment | added | EuYu | @GitGud I sort of understand the idea behind what you are proposing, but the purpose of the table is to demonstrate a bijection between $\mathbb{N}$ and $\mathbb{N}\times \mathbb{N}$. What is the purpose of your infinite extension? | |
| Aug 2, 2013 at 10:59 | comment | added | Git Gud | @EuYu Read my comment, please. | |
| Aug 2, 2013 at 10:59 | comment | added | EuYu | @HaithamGad The key point is that it's a countably infinite table. Take a look at martini's example of $1/3$. When do you think that shows up? | |
| Aug 2, 2013 at 10:58 | comment | added | Haitham Gad | Aren't they going to show up in the table at some point? After all, it's an infinite table. | |
| Aug 2, 2013 at 10:57 | comment | added | Git Gud | Why can't the three digit numbers come later? And then the 4 digits one, etc. An infinite table... and then the infinite strings may come after the finite ones, like $\omega +1$ or something. | |
| Aug 2, 2013 at 10:55 | comment | added | David Mitra | Or even, for example, $0.12$. | |
| Aug 2, 2013 at 10:55 | comment | added | martini | You are missing all real numbers which do not have a finite decimal expansion, your table does not even contain all rationals, for example $1/3 = 0.3333\ldots$. | |
| Aug 2, 2013 at 10:55 | comment | added | OR. | So far the table only have decimal representations of rational numbers. | |
| Aug 2, 2013 at 10:47 | history | asked | Haitham Gad | CC BY-SA 3.0 |