Timeline for Clarification about Cantor's Diagonal argument compared to Natural Numbers
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 6 at 11:15 | comment | added | user28434 | @PM2Ring, … and every natural number we can pair with itself, therefore set of natural numbers is countable | |
| May 6 at 1:59 | history | became hot network question | |||
| May 5 at 21:18 | comment | added | PM 2Ring | If a set is countable, that just means that we can pair each element of the set with a unique natural number. That is, given any element $e$, we can generate an index number $n$ for it, and the algorithm that generates $n$ is guaranteed to terminate in a finite number of steps for any $e$. | |
| May 5 at 20:57 | answer | added | Useless | timeline score: 1 | |
| May 5 at 18:55 | comment | added | Steen82 | The natural numbers are countable because they can be arranged in an infinite list, for example 1,2,3,4,..... or 2,1,4,3,6,5...., to give a couple of examples. Cantor's argument shows that any list of real numbers must miss at least one real number. | |
| May 5 at 18:45 | answer | added | JeffJo | timeline score: 3 | |
| May 5 at 18:39 | answer | added | Cristof012 | timeline score: 2 | |
| May 5 at 18:38 | answer | added | Paul Tanenbaum | timeline score: 4 | |
| May 5 at 18:37 | answer | added | ryan mcbean | timeline score: 2 | |
| May 5 at 18:31 | comment | added | Karl | If you try to run Cantor's argument on the natural numbers, you will get an infinite string of digits, and this is not a natural number. Even though there are infinitely many natural numbers, each one of them is represented as a finite string of digits. This is not the case for real numbers. | |
| May 5 at 18:28 | review | Close votes | |||
| May 6 at 2:45 | |||||
| May 5 at 18:06 | comment | added | Soham Saha | Suppose you have many things and you start counting them one by one, pointing at something and calling out ‘1’, then, ‘2’,… If you can show that in this way everything will be counted if you go on forever, then the ‘many things’ are countable. However, if you can’t, they’re uncountable. | |
| May 5 at 18:05 | comment | added | Thomas Andrews | Natural numbers do not have infinitely many freely-chosen digits. Any digits after the decimal point makes it not a natural number, and only finitely many digits to the left of the decimal point can be non-zero. There is no natural number with infinitely many digits. | |
| May 5 at 18:02 | comment | added | Thomas Andrews | Uncountable is not the same as infinite (unending.) The natural numbers are countably practically by definition. | |
| S May 5 at 17:59 | review | First questions | |||
| May 5 at 18:18 | |||||
| S May 5 at 17:59 | history | asked | peachyoana | CC BY-SA 4.0 |