Timeline for Measure of the same set in different models of ZF
Current License: CC BY-SA 3.0
24 events
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| Aug 6, 2014 at 17:18 | comment | added | Monroe Eskew | It's easy to show by induction that Borel sets defined by a code in the intersection of the models have absolute measure. | |
| Aug 4, 2014 at 21:39 | history | edited | Andrés E. Caicedo |
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| Aug 4, 2014 at 19:05 | comment | added | Andrés E. Caicedo | @EmilJeřábek I see. Yes, everything I said is for transitive proper class models. There may be something interesting to say for $\omega$-models, but I have not thought much about it. | |
| Aug 4, 2014 at 18:55 | comment | added | Emil Jeřábek | @Andres: My point was that if $\phi$ is an arithmetic sentence, you can write $\{x\in\mathbb R:\phi\}$ explicitly as a tree of intersection and unions (corresponding to the quantifiers in $\phi$) of intervals (whose endpoints are $\Delta^0_0$-definable, and correspond to evaluation of atomic formulas in $\phi$). I’m not in office now so I can’t check Solovay’s paper, but I would be very surprised if this can’t be expressed as a Borel code. It transpired in later comments that one needs to restrict attention to nice models (in particular, with absolute $\omega$), and I guess this is one reason. | |
| Aug 4, 2014 at 17:19 | answer | added | Ashutosh | timeline score: 4 | |
| Aug 4, 2014 at 16:58 | comment | added | Shay Ben Moshe | Andres, thanks, I'll read some of it later. Asaf, I am returning to the university in October. (finally!) and you made me hungry too. | |
| Aug 4, 2014 at 16:54 | comment | added | Asaf Karagila♦ | And I am too hungry and otherwise busy for writing an answer right now. If you're around the university tomorrow we can meet afternoon, or on Wednesday morning and talk about this. | |
| Aug 4, 2014 at 16:48 | comment | added | Andrés E. Caicedo | Shay, I am writing from my phone, and suffering with auto-correct, so no chance for a while of writing an answer, but the references I suggested should give you all the details you need. | |
| Aug 4, 2014 at 16:46 | comment | added | Andrés E. Caicedo | (@Emil The point being that the code is simply a description of the construction of the set: Start with precisely these intervals with rational end points, take their union, form the complement of the result, and so on.) | |
| Aug 4, 2014 at 16:45 | comment | added | Shay Ben Moshe | Gerald, if you have anything interesting to say in that case I would love to hear that. I have edited my question accordingly... | |
| Aug 4, 2014 at 16:44 | history | edited | Shay Ben Moshe | CC BY-SA 3.0 |
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| Aug 4, 2014 at 16:42 | comment | added | Andrés E. Caicedo | @Emil I assume your comment was for me? No, Borel codes are more robust than this. One can see a quick description in Solovay's paper on all sets of reals being measurable. These codes generalize to projections of trees, and ultimately to universally Baire sets and the setting of term relations. | |
| Aug 4, 2014 at 16:42 | comment | added | Shay Ben Moshe | Asaf, I saw the links you sent, thanks, Also, Asaf and Andres, I would love it if you could elaborate more (and maybe answer rather than comment). | |
| Aug 4, 2014 at 16:41 | comment | added | Gerald Edgar | Do you want ZFC or something, to make sure Lebesgue measure is not trivial? ... What happens if, say, the real line is a countable union of countable sets? | |
| Aug 4, 2014 at 16:38 | comment | added | Shay Ben Moshe | Asaf, the language of set theory. Also the models should probably be well-founded or something similar. | |
| Aug 4, 2014 at 16:38 | comment | added | Andrés E. Caicedo | The strongest results we have require that we assume large cardinals, and restrict our attention to models and their forcing extensions, the codes being what we call term relations. You can read about this in the Feng-Magidor-Woodin paper on universally Bare sets, and in Steel's first paper on the derived model theorem. This is now part of the core model induction machinery. | |
| Aug 4, 2014 at 16:36 | comment | added | Asaf Karagila♦ | Here is one and here is another (which might be less duplicate, but also relevant). | |
| Aug 4, 2014 at 16:35 | comment | added | Emil Jeřábek | If you replace CH with Con(ZF) in the usual counterexample, wouldn’t it have a perfectly robust code as a closed set? | |
| Aug 4, 2014 at 16:34 | comment | added | Asaf Karagila♦ | Shay, definable in what language and so on, that's the question. If you really just want to ask about changing the measure, then this is in fact a duplicate of at least two questions. Let me find links. | |
| Aug 4, 2014 at 16:33 | comment | added | Andrés E. Caicedo | Not in the way you stated it. The usual counterexample is to define $ A $ to be empty or the whole line depending on whether $\mathsf {CH} $ holds. What you want instead is to have a "robust" description, in which case the answer is yes. The problem is to formalize this "robustness". For Borel sets and their continuous images, we can exhibit "codes" that describe how the set is made up starting with basic open sets. Beyond this, we do not have entirely satisfactory codes and the problem becomes set-theoretic. | |
| Aug 4, 2014 at 16:32 | comment | added | Shay Ben Moshe | I meant in the sense of first-order formula with one free variable that defines $A$ as a subset of $\mathbb{R}$. I actually don't care about the definability, but I can't see how to ask about $A$ in different models otherwise. | |
| Aug 4, 2014 at 16:30 | comment | added | Asaf Karagila♦ | What exactly do you mean "definable" here? | |
| Aug 4, 2014 at 16:27 | review | First posts | |||
| Aug 4, 2014 at 18:01 | |||||
| Aug 4, 2014 at 16:24 | history | asked | Shay Ben Moshe | CC BY-SA 3.0 |