Limit Filters and Dependent Choice in Countable-Support Symmetric Iterations

Abstract

We isolate an algebraic limit-filter construction for countable-support symmetric iterations built from coherent successor-stage symmetric systems. At limits of uncountable cofinality, countably many generators and their conjugating automorphisms are bounded below the limit; at limits of cofinality $\omega$, countable-intersection closure is imposed in the definition. The resulting limit filters are normal and ${\omega }_1$-complete. This gives a ground-coded syntactic closure result: the canonical tuple of a ground-model countable sequence of hereditarily symmetric names is hereditarily symmetric. We also recall the standard symmetric-extension theorem yielding $\mathrm{ZF}$ from a genuine symmetric system. The previously claimed preservation of $\mathrm{DC}$ and the unordered-pairs application are withdrawn. Filter completeness alone does not turn the ground-coded tuple statement into closure under arbitrary countable sequences in the forcing extension. Moreover, every nonempty finite set of reals has a lexicographically least element in $\mathrm{ZF}$, so a family of two-element sets of reals without a choice function cannot exist. The present paper retains only the corrected limit-filter observations and the explicitly qualified symmetric-name consequences.

arXiv commentMajor correction. The previously claimed preservation of Dependent Choice and the unordered-pairs application are withdrawn. Filter completeness yields closure for ground-model coded families of hereditarily symmetric names, not arbitrary countable sequences in the extension. The corrected manuscript retains the valid limit-filter algebra and qualified symmetric-name results