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

TL;DR

This paper introduces a comprehensive framework for iterated symmetric extensions with various support sizes, enabling the construction of models of ZF with controlled choice principles and adding reals without collapsing cardinals.

Contribution

It develops a unified approach to symmetric iterations with countable and smaller support, extending to class-length iterations and handling singular cardinals with new fusion techniques.

Findings

Constructs models of ZF with added reals and no AC

Preserves ZF and certain choice principles under complex iterations

Introduces new fusion methods for singular cardinals

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 $\lambda<\kappa$.