Timeline for capturing small sets in small factors
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Dec 23, 2015 at 20:24 | comment | added | Sean Cox | Regarding Monroe's question in the comment above ("Can you determine if it's consistent that there is an inaccessible κ with this property that is not weakly compact?"): (A) if one replaces $\kappa$-cc with $\kappa$-Knaster, the answer is yes (see Thm 1.14 of arxiv.org/abs/1508.03831). (B) If every $\kappa$-cc partial order has stationarily many regular suborders of size $<\kappa$---which implies the capturing property---then $\kappa$ must be weakly compact (see Theorem 1.8 of arxiv.org/abs/1508.03831). I don't know what happens if we replace "stationarily" with "cofinally". | |
| Jun 20, 2014 at 15:50 | comment | added | Monroe Eskew | ams.org/mathscinet-getitem?mr=781072 The very last paragraph notes that it is consistent that $2^\omega = \omega_2$ while the measure algebra has density $\omega_1$. | |
| Jun 20, 2014 at 15:33 | comment | added | Yair Hayut | Yes, you're right. I thought that you want to consider only the cases where the size of the poset $\mathbb{P}$ is $\kappa$, but indeed your question is less restrictive. I think that if you do add this restriction, the rest of the answer will still be OK but it is consistent that $\omega_1$ will fulfill your requirement vacuously, i.e. there is a model of $\neg CH$ in which every c.c.c. forcing that doesn't add a Cohen real is of cardinality at least $2^{\aleph_0}$. | |
| Jun 20, 2014 at 15:10 | comment | added | Monroe Eskew | I think it's independent what the density is, I will try to find the citation. But anyway it answers the question b/c it's $\aleph_1$-c.c. and adds no Cohen reals. | |
| Jun 20, 2014 at 14:57 | comment | added | Yair Hayut | Doesn't the random real forcing have density of $2^{\aleph_0}$? | |
| Jun 20, 2014 at 14:45 | comment | added | Monroe Eskew | $\omega_1$ is already ruled out by random forcing, and almost-disjoint coding. | |
| Jun 20, 2014 at 14:42 | vote | accept | Monroe Eskew | ||
| Jun 20, 2014 at 9:58 | comment | added | saf | Everything you always wanted to know about the productivity chain condition, but were afraid to ask: assafrinot.com/paper/18 | |
| Jun 20, 2014 at 7:23 | history | edited | Yair Hayut | CC BY-SA 3.0 |
maybe omega_1 can have this property
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| Jun 20, 2014 at 7:00 | comment | added | Yair Hayut | Also, this argument doesn't rule out the possibility that $\kappa = \omega_1$ (since c.c.c. is productive under $MA$). | |
| Jun 20, 2014 at 5:44 | comment | added | Yair Hayut | I don't know. It's a very interesting question. | |
| Jun 20, 2014 at 1:56 | comment | added | Monroe Eskew | Great! Can you determine if it's consistent that there is an inaccessible $\kappa$ with this property that is not weakly compact? | |
| Jun 19, 2014 at 17:44 | history | answered | Yair Hayut | CC BY-SA 3.0 |