Symmetric Iterations with Countable and $<κ$-Support: A Framework for Choiceless ZF Extensions
Frank Gilson
TL;DR
This work develops a comprehensive, uniform framework for symmetric iterations in ZF without the Axiom of Choice, covering both countable and <κ-support. It constructs normal, κ-complete limit filters and hereditarily symmetric names to ensure ZF is preserved, while enabling controlled DC fragments through a κ-Baire or Localization hypothesis. The methodology handles set-length and class-length iterations via a unified template, and extends to singular κ using block-stabilizers and tree fusion to achieve singular-limit completeness and DC_{<κ} without collapsing κ or adding bounded subsets. The authors illustrate the framework with Cohen and Random examples, compare it to classical symmetric models, and develop a broad suite of tools (trees of conditions, game-guided fusion, block-partition stabilizers) for constructing choiceless inner models with targeted choice fragments. They also outline advanced applications, including Solovay-style regularity and measurability questions in choiceless contexts, supported by a robust consistency-strength landscape.
Abstract
We present a unified framework for symmetric iterations with countable and, more generally, $<κ$-support. Set-length iterations are handled uniformly, and, when the template is first-order definable over a Gödel-Bernays set theory with Global Choice ground, the same scheme yields class-length iterations. Limit stages with $\mathrm{cf}(λ)\geκ$ are treated by direct limits; limits with $\mathrm{cf}(λ)<κ$ are presented as inverse limits via trees of conditions and tuple-stabilizer filters. The induced limit filters are normal and $κ$-complete, which ensures closure of hereditarily symmetric names and preservation of ZF; under a $κ$-Baire (strategic-closure) hypothesis we obtain $DC_{<κ}$, and under a Localization hypothesis we obtain $DC_κ$. For countable support we give an $ω_1$-length construction that adds reals and refutes AC while preserving ZF+DC, and we show that mixed products (e.g., Cohen with Random) fit naturally via stable pushforwards and restrictions. For singular $κ$, we prove the case $\mathrm{cf}(κ)=ω$ in full using block-partition stabilizers and trees; for arbitrary singular $κ$ we introduce game-guided fusion of length $\mathrm{cf}(κ)$ and a tree-fusion master condition, yielding singular-limit completeness, preservation of $DC_{<κ}$, no collapse of $κ$, and no new bounded subsets of $κ$. The resulting toolkit provides reusable patterns for constructing choiceless inner models that retain targeted fragments of Dependent Choice.