Timeline for Can we collapse $\omega_1$ to $\omega$ without adding a dominating real?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Oct 13, 2011 at 19:50 | comment | added | Noah Schweber | (In my comment, trees grow upwards.) | |
| Oct 13, 2011 at 19:50 | comment | added | Noah Schweber | Well, as long as I'm on a roll saying false things: if there were a forcing that collapsed $\omega_1$ without adding an escaping real, the best candidate I can think of would be an uncountable analog of Sacks forcing: conditions are subtrees of $\omega_1^{<\omega}$ with every node lying below an uncountable antichain. Forcing with these trees collapses $\omega_1$, but given that Sacks forcing is $\omega^\omega$-bounding (mathoverflow.net/questions/46770/…), I would not be too shocked if this were, too. Thoughts? | |
| Oct 13, 2011 at 12:42 | comment | added | Joel David Hamkins | My reading of 26.8 has the converse sense, namely, that every forcing notion of size at most $\lambda$ embeds into the collapse of $\lambda$, which doesn't help with your claim. Nevertheless, under CH it is true that any partial order collapsing $\omega_1$ adds a Cohen real and hence an escaping real, since in this case the number of dense subsets for Cohen real forcing has become countable, and so one can construct a $V$-generic Cohen real directly. I am unsure about the general claim that every forcing notion collapsing $\omega_1$ adds a generic for $\text{Coll}(\omega,\omega_1)$. Hmmmmnn... | |
| Oct 13, 2011 at 1:25 | comment | added | Noah Schweber | smacks head Of course. One more question: what you just said, together with the fact that we can embed the usual collapse into any forcing which collapses $\omega_1$ (see Jech Third Millenium Edition, Cor. 26.8, pg. 516 for exact statement) implies that any forcing which collapses $\omega_1$ adds an escaping real, correct? | |
| Oct 12, 2011 at 22:17 | comment | added | Joel David Hamkins | Noah, that would be too much, since the collapse forcing adds a Cohen real and hence an escaping real, regardless of the value of the dominating number. And Todd has apparently explained some of the details of this in his answer. | |
| Oct 12, 2011 at 20:20 | comment | added | Noah Schweber | A quick question: it looks like your basic argument above can prove the stronger result that if the dominating number is large, then the usual collapse does not add an escaping real, that is, a real that is not dominated by any function in the ground model. Is this correct? | |
| Oct 12, 2011 at 16:45 | vote | accept | Noah Schweber | ||
| Oct 12, 2011 at 16:45 | comment | added | Noah Schweber | This is an absolutely fantastic answer! Thanks. | |
| Oct 12, 2011 at 14:38 | comment | added | Joel David Hamkins | Yes, now fixed. | |
| Oct 12, 2011 at 14:37 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Oct 12, 2011 at 14:33 | comment | added | Amit Kumar Gupta | Hey Joel, I think the first paragraph of your proof should say "1 implies 2" | |
| Oct 12, 2011 at 13:27 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 193 characters in body; added 136 characters in body
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| Oct 12, 2011 at 13:25 | comment | added | Todd Eisworth | Slight generalization: No notion of forcing of size less than $\mathfrak{d}$ can add a dominating real. (But Joel got to it first! ;) ) | |
| Oct 12, 2011 at 13:18 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |