Symmetric Iterations with Countable Support

Abstract: We develop a framework for iterated symmetric extensions with countable support. At limit stages of uncountable cofinality we use the direct-limit presentation; at limits of cofinality $ω$ we use a countable-support (inverse-limit) presentation via trees of conditions along a fixed cofinal sequence. We define the associated limit symmetry filter by head pushforwards and then (only at cf$(λ)=ω)$ closing under countable intersections in the minimal normal way, and we prove that the resulting limit filter is normal and $ω_1$--complete. This yields closure of hereditarily symmetric names under countable tuples and, consequently, preservation of ZF; in a ZFC ground model we also obtain DC in the resulting symmetric model. When the iteration template is first-order definable over a GBC ground with Global Choice and sufficient class-recursion, the same scheme extends to class-length iterations, with the final symmetric model obtained from class hereditarily symmetric names (rather than as a union of stage models).