Symmetric Iterations with Countable Support
Frank Gilson
TL;DR
This work addresses constructing models of $ZF$ in which the Axiom of Choice fails by developing a uniform framework for iterated symmetric extensions with countable support. The core idea combines direct-limit presentations at uncountable cofinalities and inverse-limit (tree) presentations at cofinality $\omega$, producing a normal $\omega_1$-complete limit filter that closes hereditarily symmetric names under countable tuples, thereby preserving $ZF$ (and, from a $ZFC$ ground, $DC$). The framework also extends to class-length iterations under definability assumptions, offering a reusable toolkit for building choiceless models with Dependent Choice. Collectively, this advances the methodology beyond finite-support approaches, enabling controlled failures of choice with robust foundational properties and broad applicability to set-length and class-length constructions.
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).