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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