Timeline for Independent families on $\omega$ with an additional splitting property
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Dec 3, 2021 at 15:31 | answer | added | Jonathan Schilhan | timeline score: 5 | |
| Dec 3, 2021 at 14:14 | comment | added | Paolo Leonetti | @JoelDavidHamkins Thanks for your comments, Joel and Will. Since I am not expert in this field, could you provide me some more details of your ideas? | |
| Nov 30, 2021 at 13:34 | comment | added | Joel David Hamkins | @WillBrian, I think if you are careful about how much genericity you need in those arguments, you will get something like the construction I describe in my comment. | |
| Nov 30, 2021 at 13:26 | comment | added | Will Brian | I don't know if you're interested in consistency results, but this is certainly true in the Cohen model. Your independent family can consist of $\mathfrak{c}$ mutually generic Cohen reals, and your bijection can map each real in $\Omega$ to something Cohen-generic with respect to that real (which is all but countably many of the Cohen reals in your independent family). The same thing works with random reals. | |
| Nov 30, 2021 at 11:57 | comment | added | Joel David Hamkins | Can't one provide an affirmative answer by growing the sets $F(X)$ continuously based on finite approximations to $X$? That is, as we learn more about $X$, when a lot of new elements are added we add one of these to $F(X)$ and when a lot of new elements are omitted from $X$ we add one of these to $F(X)$. In the limit, we will thereby achieve the desired infinite property. And similarly we ensure independence by adding new elements periodically for each Boolean expression of combining them. In this way, we grow the set $F(X)$ continuously from $X$ with your desired properties. | |
| Nov 30, 2021 at 11:26 | history | edited | Paolo Leonetti | CC BY-SA 4.0 |
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| Nov 30, 2021 at 11:20 | history | edited | Paolo Leonetti | CC BY-SA 4.0 |
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| Nov 30, 2021 at 11:15 | comment | added | Paolo Leonetti | @JoelDavidHamkins Thank you for the correction | |
| Nov 30, 2021 at 11:14 | history | edited | Paolo Leonetti | CC BY-SA 4.0 |
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| Nov 30, 2021 at 11:13 | comment | added | Joel David Hamkins | In your question, you say $\forall X\subseteq\omega$, but I guess you intend only infinite/coinfinite $X$? Otherwise, it would clearly be impossible. | |
| Nov 30, 2021 at 11:08 | history | edited | Paolo Leonetti | CC BY-SA 4.0 |
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| Nov 30, 2021 at 10:54 | history | asked | Paolo Leonetti | CC BY-SA 4.0 |