# logic – Symmetric Product Forcing

Given a family of forcing notions $$(P_i)_{iin I}$$ we can take the product $$P:=prod_{iin I}P_i$$ as a forcing notion to create a generic filter of the form $$G=(G_i)_{iin I}$$ such that for each $$iin I$$ the projection $$G_i$$ corresponds to the generic filter created when forcing with $$P_i$$. This is called product forcing and allows us to adjoin several different types of generic objects at once. (For a more detailed discussion of the subject see Product forcing and generic objects)

Now my question is if and how product forcing can be combined with symmetric forcing. Assume we have a family of forcing notions as above and a family of groups $$(mathcal{G}_i)_{iin I}$$ as well as $$(mathcal{F}_i)_{iin I}$$ such that $$mathcal{G}_i$$ is a subgroup of $$Aut(P_i)$$ and $$mathcal{F}_i$$ is a normal filter on $$mathcal{G}_i$$ for all $$iin I$$. Can we just define $$P$$ as above with $$mathcal{G}:=prod_{iin I}mathcal{G}_i$$ acting on $$P$$ componentwise and $$mathcal{F}simeqprod_{iin I}mathcal{F}_i$$ as a normal filter on $$mathcal{G}$$ ?

E.g. consider Cohen’s original symmetric model of $$ZF+neg AC$$ where he adjoins countably many generic reals and then proceeds to construct an infinite subset $$Asubset mathbb{R}$$ without any countably infinite subsets. Then the construction described above should allow us to adjoin $$I$$ many such sets $$(A_i)_{iin I}$$ at once.

Are there any complications one might encounter of with this type construction (i.e. symmetric product forcing)? Is there any literature on the subject?