Symmetric Iterations with Countable and $<\kappa$-Support: A Framework for Choiceless ZF Extensions

Abstract

We develop a unified framework for iterated symmetric extensions with countable support and, more generally, with $<\kappa$-support. Set-length iterations are treated uniformly, and when the iteration template is first-order definable over a Godel-Bernays ground with Global Choice, the construction extends to class-length iterations. At limit stages with $\mathrm{cf}(\lambda)\ge\kappa$ we use direct limits; when $\mathrm{cf}(\lambda)<\kappa$ we use inverse-limit presentations via trees of conditions together with tuple-stabilizer symmetry filters. The resulting limit filters are normal and $\kappa$-complete, yielding closure of hereditarily symmetric names and preservation of $\mathrm{ZF}$. Under a $\kappa$-Baire (strategic closure) hypothesis we obtain $DC_{<\kappa}$, and under a Localization hypothesis we obtain $DC_\kappa$. In the countable-support setting we give an $\omega_1$-length construction adding reals and refuting $\mathrm{AC}$ while preserving $\mathrm{ZF}+\mathrm{DC}$, and we treat mixed products via stable pushforwards and restrictions. For singular $\kappa$, we develop the $\mathrm{cf}(\kappa)=\omega$ case using block-partition stabilizers and trees; for arbitrary singular $\kappa$ we introduce game-guided fusion of length $\mathrm{cf}(\kappa)$ and a tree-fusion master condition, obtaining singular-limit completeness, preservation of $DC_{<\kappa}$, no collapse of $\kappa$, and no new subsets of any $\lambda<\kappa$.