Timeline for Can a generic $\mathbb{R}$ have a new cardinality?
Current License: CC BY-SA 4.0
16 events
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| S Aug 12, 2020 at 23:04 | history | bounty ended | CommunityBot | ||
| S Aug 12, 2020 at 23:04 | history | notice removed | CommunityBot | ||
| Aug 9, 2020 at 4:25 | comment | added | Noah Schweber | @DryBones No it is not, at least to me. There are things one might try on the general grounds that they're usually relatively easy and instructive - e.g. Cohen forcing - but nothing stands out. | |
| Aug 9, 2020 at 4:02 | comment | added | Dry Bones | Thanks, as expected your answer gives me a lot more to learn about! But if I got it partially, then it's not clear how $V\models{\sf AD}$ might help to choose $\Bbb P$ and $G$ with the desired properties, is that right? | |
| Aug 9, 2020 at 3:48 | comment | added | Noah Schweber | Moreover, determinacy models show up as natural submodels ("inner models") of choice models - e.g. assuming large cardinals, $L(\mathbb{R})$ satisfies $\mathsf{ZF+DC+AD}$. In a sense, determinacy is how choice fails "canonically." So looking at models of $\mathsf{ZF+AD}$ (or its ilk) is a natural thing to do here. Unfortunately, the question of how much we can conclude about $V[G]$ from knowing that $V\models\mathsf{AD}$ is a complicated one. In particular, unlike choice determinacy is not preserved by forcing. Forcing over $\mathsf{AD}$-models is as far as I'm aware still not well-understood. | |
| Aug 9, 2020 at 3:44 | comment | added | Noah Schweber | @DryBones The construction of $V[G]$ makes sense as long as $V\models\mathsf{ZF}$ - or indeed vastly less ($\mathsf{KP}$ is more than enough). Per the end of my question, I know the answer - for silly reasons - if we consider $V\models\mathsf{ZFC}$. Meanwhile, I know basically nothing about what happens if we consider $V\models \mathsf{ZF+\neg AC}$. So why $\mathsf{AD}$? Well, many choiceless models are truly terrible (e.g. the reals can be a countable union of countable sets); amongst the $\mathsf{ZF+\neg AC}$-models, the $\mathsf{AD}$-models are in many ways very nicely behaved. (cont'd) | |
| Aug 9, 2020 at 3:33 | comment | added | Dry Bones | I'm taking a look at Kunen's book and I like it! Though I'm still far from properly 'understanding' your question in detail, I'm compelled to ask if you know the answer for, say, ${\sf ZF}+\lnot{\sf AD}$. I mean, should the construction of $V[G]$ rely on some property of $V$ that depends on $V\models{\sf AD}$ ? Maybe this question doesn't make sense, but showing me why will probably help a bit on finding my way through this labyrinth! =] | |
| Aug 5, 2020 at 19:41 | comment | added | Noah Schweber | @DryBones To be fair I might be overemphasizing the role of model theory. It's not really directly needed if you're willing to take a couple things on faith. However, the basic idea behind forcing is that we're building models of a certain very complicated theory (namely $\mathsf{ZF}$ or similar), and understanding basic properties of models will help dispel many (very reasonable) initial worries. So personally I wouldn't approach forcing without being comfortable with the downward Lowenheim-Skolem, compactness, completeness, and incompleteness theorems - the experience that confers will help. | |
| Aug 5, 2020 at 19:38 | comment | added | Dry Bones | Ok, I can imagine. Thanks, anyway! | |
| Aug 5, 2020 at 19:36 | comment | added | Noah Schweber | Forcing, however, relies on a mastery of the basics of set theory and model theory. So it's quite a long road. | |
| Aug 5, 2020 at 19:36 | comment | added | Noah Schweber | @DryBones Unfortunately this really isn't accessible without a significant amount of experience - specifically, comfort with forcing is the key ingredient (although you also need to be comfortable with the axiom of determinacy, that's somewhat a secondary concern - the question is still interesting if we replace $\mathsf{ZF+AD}$ with just $\mathsf{ZF}$). Forcing is treated for example in Kunen's old book "Set theory and independence proofs;" this and this are also good. | |
| Aug 5, 2020 at 19:32 | comment | added | Dry Bones | What reference would you indicate for a beginner in Logic willing to understand what is being asked? =] | |
| S Aug 4, 2020 at 21:38 | history | bounty started | Noah Schweber | ||
| S Aug 4, 2020 at 21:38 | history | notice added | Noah Schweber | Draw attention | |
| Aug 2, 2020 at 6:54 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| Aug 2, 2020 at 6:49 | history | asked | Noah Schweber | CC BY-SA 4.0 |