Timeline for Well-ordered cofinal subsets
Current License: CC BY-SA 3.0
10 events
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| Dec 5, 2011 at 13:19 | comment | added | Carl Mummert | As a side remark, the fact that a general poset won't have cofinal linearly ordered subsets is one motivation for looking at filters (or ideals) on partial orders rather than linearly ordered subsets. For example, every maximal filter $F$ on a poset is cofinal in the sense that there is no element of the poset strictly below all elements of $F$. | |
| Dec 4, 2011 at 23:09 | comment | added | Asaf Karagila♦ | Finding a somewhat newer reference, Paul Howard and Hartmut Höft show in projecteuclid.org/euclid.ndjfl/1040511347 some results regarding this principle, while in ZFA it is rather understood, there seem to me that some questions remain open regarding its provability in ZF. While no final word is said about whether or not CF implies AC or not; if they only conjecture that "Well orders have no decreasing chains" do not imply CF over ZF, then I'd assume they could not find a good reference for the fact that CF implies AC; nor they mention otherwise. | |
| Dec 4, 2011 at 20:05 | comment | added | Asaf Karagila♦ | Jech's Axiom of Choice has this as a ** exercise with no hints. It refers to a paper by Morris, but the footnote explains that Felgner claimed otherwise (that the Cofinality Principle implies AC). Morris found a mistake in Felgner's argument which he supposedly corrected. No further information is given on the topic. | |
| Dec 4, 2011 at 19:57 | comment | added | Joel David Hamkins | Oh, of course! But what about the other claim, that it is not equivalent to AC? | |
| Dec 4, 2011 at 19:52 | comment | added | Asaf Karagila♦ | @Joel: I mentioned Cohen's basic model in my answer. The Dedekind-finite set of reals is linearly ordered but has no cofinal well-ordered subset. | |
| Dec 4, 2011 at 19:51 | comment | added | Joel David Hamkins | Thanks, Asaf. So you are saying that it is a strictly weak choice principle, then? What are the models separating it from ZF and also from ZFC? | |
| Dec 4, 2011 at 19:48 | comment | added | Asaf Karagila♦ | The assertions that every linearly ordered set has a cofinal well ordered subset, along with the order extension principle (every poset can be extended into a linear order) imply the axiom of choice. Each does not imply the other, though. | |
| Dec 4, 2011 at 19:47 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 529 characters in body; added 43 characters in body
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| Dec 4, 2011 at 19:42 | vote | accept | Habujew | ||
| Dec 4, 2011 at 19:41 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |