Timeline for Models of ZF intermediate between a model of ZFC and a generic extension
Current License: CC BY-SA 4.0
7 events
when toggle format | what | by | license | comment | |
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Jan 20 at 10:47 | comment | added | Asaf Karagila♦ | @Lorenzo Send me an email. This question has a long answer. | |
Jan 20 at 10:30 | comment | added | Lorenzo | @AsafKaragila I have a question. In Grigorieff's Intermediate submodels and generic extensions in set theory he proves that every symmetric submodel of $M[G]$ is of the form $(\mathsf{HOD} (M(x)))^{M[G]}$ for some $x \in M[G]$ (Theorem C), and then he observes that every inner model of this kind is equal to $M(y)$ for some $y \in M[G]$ with $y\subset M$ (Corollary 2, Section 9). Thus every $M(y)$ for some $y \in M[G]$ with $y\subset M$ is a symmetric extension of $M$. The novelty of Usuba's result is that $M(y)$ is a symmetric extension of $M$ even when $y$ is not a subset of $M$? | |
Jan 23, 2020 at 19:44 | comment | added | Asaf Karagila♦ | But regardless, feel free to drop me an email if you have any followups or any other choiceless questions... | |
Jan 23, 2020 at 19:32 | comment | added | Asaf Karagila♦ | The point is that $L(V_\alpha^M)$ where $M$ is the Bristol model are all symmetric models. | |
Jan 23, 2020 at 19:25 | vote | accept | Toby Meadows | ||
Jan 23, 2020 at 19:22 | comment | added | Toby Meadows | The Bristol model is pretty intimidating. Can you elaborate a little on your second comment? Happy to move to email if that is preferable. | |
Jan 23, 2020 at 19:19 | history | answered | Asaf Karagila♦ | CC BY-SA 4.0 |