Timeline for Order types of positive reals
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 7, 2013 at 13:49 | comment | added | Joel David Hamkins | This argument is due originally to Cantor, and is the origin of the back-and-forth method used throughout model theory. As elsewhere, the universal property here uses only "forth". | |
| Nov 1, 2010 at 15:48 | comment | added | Pace Nielsen | Ah, that wasn't clear from your original answer. Now that you say that, it makes total sense. | |
| Oct 31, 2010 at 7:44 | comment | added | Robin Chapman | No, I have not altered the order type. All I have done is use a bijection between $\mathbb{N}$ and the ordered set. In $\omega+1$, $\omega$ is still the largest element -- but it's the one you insert first. So in your example we could start by mapping $\omega$ to $0$, then map $0$ to $-1$, then $1$ to $-1/2$ etc. The insight is that one doesn't have to insert your elements in increasing order! | |
| Oct 30, 2010 at 14:44 | comment | added | Pace Nielsen | But the original question asked about all possible countable order types. If you re-order things, you have changed the order type. For example, if you rearrange the countable set $\omega+1$ as $\omega,0,1,2,\ldots$, your new order type is just $\omega$. | |
| Oct 30, 2010 at 6:59 | comment | added | Robin Chapman | It does work: $\omega+1$ is a countable set. We can write its elements as $\omega,0,1,2,\ldots$. This is an order we can insert them in. There is no "final" entry. | |
| Oct 29, 2010 at 19:53 | comment | added | Pace Nielsen | I don't get this answer. The ordinal $\omega+1$ is an order-type that is countable, but your method of embedding won't work without some modification (if you place a_i at the rational number i, there will be no room left for the final entry). I would recommend David checkbox gowers answer, as it actually covers all of the countable well-orderings. | |
| May 18, 2010 at 15:41 | vote | accept | David Eppstein | ||
| May 18, 2010 at 8:16 | history | edited | Robin Chapman | CC BY-SA 2.5 |
spelling correction and improved formatting
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| May 18, 2010 at 7:36 | history | answered | Robin Chapman | CC BY-SA 2.5 |