Can we have a Choice-AntiChoice chain?

Question

Can we have a $\kappa$ sized sequence $\mathcal S$ of transitive domains $\mathcal M_i$ of models of $``\text{ZF + negation of choice}"$, where $\kappa$ is inaccessible, such that for all $i < \kappa$ we have $\mathcal M_i \in \mathcal M_{i+1}$ and such that for each $\mathcal M_i$ choice over$\mathcal M_i$ applies in $\mathcal M_{i+1}$ , i.e., for each element $s$ of $\mathcal M_i$ there exists a set $c \in \mathcal M_{i+1}$ such that $c$ has exactly one element from each element of $s$ among its elements.

Answer 1

I believe the following works (and it's not even an iterated forcing argument):

Fix $\kappa$ inaccessible . For a worldly cardinal $\mu<\kappa$, let $\mathbb{P}_\mu$ be the forcing consisting of maps from $\mu$ to $2$ with domain of cardinality $<\mu$. Let $(\mu_\eta)_{\eta<\kappa}$ enumerate the worldly cardinals $<\kappa$ in increasing order (recalling that every inaccessible cardinal is a limit of worldly cardinals), and consider the forcing notion $$\mathbb{P}=\{P\in\prod_{\eta<\kappa}\mathbb{P}_{\mu_\eta}: \vert\{\alpha<\kappa: P(\alpha)\not=\emptyset\}\vert<\kappa\}$$ be the $<\kappa$-support product of the $\mathbb{P}_\mu$s. Note that forcing with $\mathbb{P}$ preserves the inaccessibility of $\kappa$ (think about how $\mathbb{P}$ "splits" into a $\mu$-c.c. and a $\mu$-closed piece, for every $\mu<\kappa$, and count the nice names ...).

Let $G$ be $\mathbb{P}$-generic. For $\eta<\kappa$, let $G_\eta=G\upharpoonright \mu_\eta$, and note that $G_\eta$ is set-generic over the ZFC-model $L_{\mu_{\eta+1}}$. Now by a standard symmetric submodel argument we can whip up for each $\eta$ a $W_\eta$ such that:

• $L_{\mu_{\eta+2}}\subset W_\eta\subset L_{\mu_{\eta+2}}[G_{\eta+1}],$

• $W_\eta\models$ ZF+$\neg$AC, and

• $G_\eta\in W_\eta$.

The $W_\eta$s satisfy the desired properties (the third condition is what ensures that each one contains a well-ordering of the earlier ones).

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EDIT: Deleted a broken idea.