9 papers
A One-Step Cascade Symmetric Model: Rank- Packets, Binary Shielding, and the Even Exact-Cardinality Profile
Frank Gilson
We introduce a one-step cascade symmetric system whose local symmetry geometry is organized by finite -closed windows and one-step stars rather than by rowwise-independent toggl…
Explicit separation of quadratic irrationals from the middle-third Cantor set
Frank Gilson
An earlier form of this manuscript claimed, subject to a ``shallow contribution'' hypothesis, a log-square upper bound for the first occurrence of the ternary digit in a quadra…
Limit Filters and Dependent Choice in Countable-Support Symmetric Iterations
Frank Gilson
We isolate an algebraic limit-filter construction for countable-support symmetric iterations built from coherent successor-stage symmetric systems. At limits of uncountable cofinal…
A countable-support symmetric iteration separating PP from AC
Frank Gilson
We study a countable-support Cohen symmetric seed model and a proposed symmetric-iteration approach to separating the Partition Principle from the Axiom of Choice. Th…
Arithmetical Complexity and Absoluteness of Rigidity Phenomena for Ulam Sequences
Frank Gilson
We analyse the arithmetical complexity and forcing absoluteness of natural statements about Ulam sequences. For positive integers , membership in and the increasing e…
From Internal to External: Classical Models of ZF + PP + AC
Frank Gilson
An earlier form of this manuscript claimed to obtain a classical symmetric model of by two routes: a Boolean-valued externalization of an…
Symmetric Iterations with Countable and -Support: A Framework for Choiceless ZF Extensions
Frank Gilson
This replacement corrects the earlier manuscript. Its claimed general countable- and -support iteration framework does not establish the advertised preservation or application…
Partition Principle without Choice via Symmetric Iterations and Sheaf-Toposes
Frank Gilson
An earlier form of this manuscript proposed that a transformation-groupoid topos associated with a free action of a nontrivial finite group on Cantor space contains an internal mod…
The Modal Logic of Finitely Symmetry-Preserving Iterated Extensions is Exactly S4
Frank Gilson
We determine the ZF-provable modal logic of the modality , where means ' holds in every finite symmetry-preserving iteration' of the…