Timeline for Probabilities independent of ZFC?
Current License: CC BY-SA 2.5
14 events
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| Mar 9, 2011 at 16:34 | comment | added | Andreas Blass | Joel, I'd expect that the details concerning my comment are in the Bartoszynski-Judah book "Set Theory: On the Structure of the Real Line". The case of Laver forcing is in a paper of Pawlikowski, "Laver's forcing and outer measure" in the proceedings of the 1992-94 BEST. He says the result is due to Woodin, rediscovered by Judah and Shelah, but "[Woodin's proof] is unpublished and the proof in [the Judah-Shelah paper on the Kunen-Miller chart] is somewhat difficult to follow." | |
| Mar 4, 2011 at 23:23 | comment | added | Joel David Hamkins | Gunter Fuchs and I observed that if the ground model reals are ever measurable in a forcing extension $V[G]$ with a new real, then the measure of $\mathbb{R}^V$ must be $0$, since you can perform a Vitali-like argument by considering translates of the ground model unit interval by a new real and all its powers, which must be disjoint. | |
| Mar 3, 2011 at 2:24 | comment | added | Joel David Hamkins | Thanks, Andreas, do you know where we can find the details? | |
| Mar 3, 2011 at 1:55 | comment | added | Andreas Blass |
There are models of set theory in which the set of constructible reals is not Lebesgue measurable. If I remember correctly, examples include the models obtained from $L$ by adjoining a random real, a Laver real, a Sacks real, or a Miller real. These models satisfy CH, but countable support iterations of Laver, Miller, or Sacks forcing give models where the continuum has cardinality $\aleph_2$ yet the constructible reals are again non-measurable. And you can get the same result with even larger continuum by forcing with a big measure algebra to add a lot of random reals.
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| Mar 2, 2011 at 16:25 | comment | added | user11618 | Right. Of course it is consistent that every $\Sigma^{1}_{2}$ set is measurable. I'm quite interested in this thing, because in my research i'm working with a $\Delta_{2}^{1}$ set. To work with it I need to assume something (such as Martin's Axiom + \neg CH, say) to force its measurability (MA + \negCH, implies that every $\Sigma^{1}_{2}$ set is measurable). So I'm interested to learn if there are ways to prove that it is measurable in ZFC alone, perhaps showing that it is a "constructible set" (I'm not an expert at all in this area, so i'm just guessing). | |
| Mar 2, 2011 at 13:55 | comment | added | Joel David Hamkins | The set of constructible reals is all the reals and hence Borel in $L$ itself, but in forcing extensions $L[G]$ it has complexity $\Sigma^1_2$, and I believe this bound can be realized (one must not collapse $\omega_1^L$), so it can happen that a Borel set is made non-Borel by forcing. But this doesn't fully answer my question.... | |
| Mar 1, 2011 at 20:42 | comment | added | Joel David Hamkins | Matteo, that is an interesting question. More generally, if $V\subset V[G]$ is any forcing extension, then must $\mathbb{R}^V$ be measurable in $V[G]$? Must it always have either full measure or measure $0$? Hmmmmm... | |
| Mar 1, 2011 at 19:02 | vote | accept | sebastian | ||
| Mar 1, 2011 at 15:22 | comment | added | user11618 | @Joel: I have the following naive question: is it true that the the set you defined is at least measurable in all models of set theory? Or in other words, is it possible to re-state your conclusion as "because in some models of set theory this probability is 1 and in others it is 0, and in others is not defined at all". | |
| Mar 1, 2011 at 14:06 | comment | added | Joel David Hamkins | I added a more concrete example. | |
| Mar 1, 2011 at 14:05 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
Added constructibility example
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| Mar 1, 2011 at 13:28 | comment | added | sebastian | thanks a lot. that is already in the direction I am looking for. Do you know of any non-absolute description that one could use that is still "simple"? More precisely, the example above is a non-absolute description by basically providing a direct link from the sequences to the cardinal patterns. is there something more "basic" or "elementary"? I understand that it has to be somewhat more complex than the Borel code. thx a lot again! | |
| Mar 1, 2011 at 11:52 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
added 407 characters in body; added 28 characters in body; added 14 characters in body; added 215 characters in body; added 12 characters in body
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| Mar 1, 2011 at 11:28 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |