Timeline for Two versions of "absolutely ccc"
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 9, 2012 at 12:07 | comment | added | Juris Steprans | It is worth noting that splitting the gap makes F(A,B) ccc for the same reason as collapsing $\omega_1$. After splitting the gap, the partial order for filling it has a countable dense subset. | |
| Mar 26, 2012 at 7:37 | comment | added | Joel David Hamkins | I asked a question about a similar approach, using almost special Aronszajn trees instead of Hausdorff gaps, here: mathoverflow.net/questions/92235/… | |
| Mar 26, 2012 at 6:52 | comment | added | Goldstern | Yes, it seems you are right. I have edited the post to point the GAP in my reasoning. | |
| Mar 26, 2012 at 6:50 | history | edited | Goldstern | CC BY-SA 3.0 |
pointed out a mistake in my argument
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| Mar 24, 2012 at 6:36 | comment | added | user642796 | To further Joel's point (or confusion) I believe that the usual construction of a Hausdorff gap yields an indestructible gap. It thus seems that this answer would contradict what I've stated in the question about the two formulations of absolute ccc-ness being consistently equivalent. | |
| Mar 23, 2012 at 18:39 | comment | added | Joel David Hamkins | So I am unsure the example works now, because the forcing to fill the gap must collapse $\omega_1$, which destroys the hypothesis necessary to conclude that $F(A,B)$ is not ccc. | |
| Mar 23, 2012 at 18:34 | comment | added | Joel David Hamkins | It seems that the explanation is that equivalence you mention only works provided that (A,B) remains an $(\omega_1,\omega_1)$-pre-gap, which isn't itself preserved by all forcing, although it would be preserved by $\omega_1$-preserving forcing. | |
| Mar 23, 2012 at 18:29 | comment | added | Joel David Hamkins | Martin, I'm a bit confused, because you are saying that F(A,B) is absolute and loses in the ccc in any forcing extension in which the gap is filled. But suppose that we force everything here to be countable? In that extension, the gap will be filled and the forcing will be ccc because it was made countable. So what you say can't be literally true. | |
| Mar 23, 2012 at 18:23 | history | answered | Goldstern | CC BY-SA 3.0 |