Timeline for Descending almost-contained subsets of $\omega$ [closed]
Current License: CC BY-SA 4.0
16 events
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| Dec 14, 2018 at 20:33 | history | closed |
YCor Monroe Eskew Andreas Blass Ramiro de la Vega Mark Wildon |
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| Dec 10, 2018 at 21:34 | comment | added | YCor | The corrected version seems indeed as interpreted by Nik. It simply restates as: does the poset $\mathcal{P}(\omega)/\mathrm{fin}$ have a decreasing chain of type $\omega_1$? this is actually very standard and even in this version might be more suitable for MathSE. | |
| Dec 10, 2018 at 21:28 | comment | added | Andreas Blass | Since @YCor has answered (in a comment) the question as asked and has suggested the probably intended modification to make it nontrivial, and since Nik Weaver has answered the nontrivial version, I'm voting to close. (I'd favor reopening if (and only if) a different nontrivial version were proposed.) | |
| Dec 10, 2018 at 18:05 | comment | added | YCor | Well, I have to insist that it's still trivial (in ZFC): for every non-successor cardinal $\alpha<2^{\aleph_0}$, choose an infinite subset $B_\alpha$, so that the chosen subsets $B_\alpha$ have pairwise infinite symmetric difference. Define $B_{\alpha+n}$ as $B_{\alpha+n-1}$ minus a point. Finally define $A_\alpha=B_\alpha\times\omega\subseteq\omega^2$. | |
| Dec 10, 2018 at 17:59 | history | edited | Forever Mozart | CC BY-SA 4.0 |
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| Dec 10, 2018 at 17:51 | history | edited | Forever Mozart | CC BY-SA 4.0 |
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| Dec 10, 2018 at 17:48 | comment | added | Asaf Karagila♦ | Obligatory keyword reference: $\frak t$, by the way. | |
| Dec 10, 2018 at 17:46 | history | edited | Forever Mozart | CC BY-SA 4.0 |
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| Dec 10, 2018 at 17:45 | comment | added | Forever Mozart | @YCor I would like the elements of the sequence to be distinct if possible | |
| Dec 10, 2018 at 17:45 | review | Close votes | |||
| Dec 14, 2018 at 20:33 | |||||
| Dec 10, 2018 at 17:43 | comment | added | YCor | As regards the current question, the constant sequence $A_\alpha=A$ works... Second, your axioms on the sequence make no relation between $A_\alpha$ and $A_\beta$ except when $\alpha$ and $\beta$ differ by addition by an integer. Please write your question more carefully. (Your last edit doesn't address my second sentence. Currently it's obvious to arrange a sequence of length $2^{\aleph_0}$.) | |
| Dec 10, 2018 at 17:42 | answer | added | Nik Weaver | timeline score: 2 | |
| Dec 10, 2018 at 17:37 | comment | added | Forever Mozart | @AsafKaragila I would like for $A_{\alpha+1}$ to be contained in $A_\alpha$, up to some finite error. | |
| Dec 10, 2018 at 17:36 | comment | added | Asaf Karagila♦ | Do you want $\subseteq^*$ or just finite differences? Because the answer is yes in either case, but for different reasons. Please settle on a version before I put any more energy into writing an answer... | |
| Dec 10, 2018 at 17:31 | history | edited | Forever Mozart | CC BY-SA 4.0 |
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| Dec 10, 2018 at 17:17 | history | asked | Forever Mozart | CC BY-SA 4.0 |