Timeline for Is the Ordering Principle equivalent to a selection principle?
Current License: CC BY-SA 4.0
13 events
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| Dec 24, 2023 at 5:01 | comment | added | Zuhair Al-Johar | Very nice and straight up to the point! This actually makes me wonder whether every statement demanding choice is equivalent to some selection principle. | |
| Dec 24, 2023 at 4:52 | vote | accept | Zuhair Al-Johar | ||
| Dec 23, 2023 at 22:32 | comment | added | Joel David Hamkins | Evidently. But the literal reading seemed wrong, since my choice principle isn't cheating like that. | |
| Dec 23, 2023 at 21:50 | comment | added | Asaf Karagila♦ | You seem to have misread my comment, Joel. :-) | |
| Dec 23, 2023 at 21:48 | comment | added | Joel David Hamkins | Regarding my remark, choice for pairs does not prove every set is linearly orderable, since then it would imply choice for finite sets, which it doesn't. | |
| Dec 23, 2023 at 20:58 | comment | added | Joel David Hamkins | In particular, in statement 2 we're just choosing a strictly larger linear order on a subset, not a linear order of the whole of $X$. | |
| Dec 23, 2023 at 20:39 | comment | added | Joel David Hamkins | But why is it convoluted? Obviously every linearly ordered proper subset of $X$ can be extended to slightly larger linearly ordered subsets, and the ordering principle is exactly equivalent to having a choice function for this. Well-orderability is equivalent to being able to do this while adding just one new point. | |
| Dec 23, 2023 at 20:36 | comment | added | Joel David Hamkins | @AsafKaragila I don't quite understand your comment. Are you saying that if we can choose from pairs, then every set is linearly orderable? (But in any case, I'm just trying to answer the question.) | |
| Dec 23, 2023 at 20:32 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Dec 23, 2023 at 20:32 | comment | added | Asaf Karagila♦ | Feels a bit convoluted, to be honest. "A set can be linearly ordered if and only if for every two points in the set we can choose a linear ordering which puts one above the other", although admittedly not as contrived. | |
| Dec 23, 2023 at 20:31 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Dec 23, 2023 at 20:16 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Dec 23, 2023 at 20:10 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |