arXiv subjects

Frank Gilson

Publications and source records attributed to Frank Gilson.

9 recordsLinked to original sources

arXiv2026

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

This replacement corrects the earlier manuscript. Its claimed general countable- and $<κ$-support iteration framework does not establish the advertised preservation or application theorems. The proposed limit filters can be improper; the dependent-choice arguments pass incorrectly from ground-model sequences of hereditarily symmetric names to arbitrary sequences in a generic extension; the class-length union argument does not verify Separation, Replacement, or Power Set; the Partition Principle argument uses an invalid same-generic compatibility inference; and the singular-support fusion argument assumes that a coherent inverse-limit family is a condition even when its support may have size $κ$. Accordingly, the main preservation, construction, class-length, dependent-choice, Partition Principle, and singular-cardinal claims are withdrawn. We retain only several conditional algebraic observations: pullback and completion operations on subgroup filters, closure of ground-model families of hereditarily symmetric names under canonical tupling, the standard ZF theorem for a genuine symmetric system, and ordinary conditional tail-closure facts. The remainder is preserved as a historical record of the superseded approach and is not asserted.

math.LO↗

arXiv2026

Partition Principle without Choice via Symmetric Iterations and Sheaf-Toposes

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 model of $\mathsf{IZF}+\mathsf{PP}+\neg\mathsf{AC}$, with a classical upgrade after double-negation sheafification. That construction is not valid. The proposed local embedding argument does not establish the required property for general epimorphisms; the finite-fiber class of small maps does not support an $\mathsf{IZF}$-universe because it does not contain the natural numbers object; the quotient map used to witness failure of choice is not a nonsplitting epimorphism in the asserted form; and double-negation sheafification does not repair these defects. The symmetric-iteration appendix also used an invalid same-condition equivariance inference. Accordingly, this replacement withdraws all model-existence, preservation, and independence claims. We retain the conditional categorical observation that a genuinely matching local family of embeddings descends to a global embedding, and record the precise obstructions so that the withdrawn construction is not cited as a solution of the Partition Principle problem.

math.LO↗

arXiv2026

From Internal to External: Classical Models of ZF + PP + $\neg$AC

An earlier form of this manuscript claimed to obtain a classical symmetric model of $\mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$ by two routes: a Boolean-valued externalization of an internal sheaf-topos construction, and a direct symmetric extension based on $\operatorname{Fn}(\mathbb{N}\times H,2)$. Those claims are withdrawn. The decisive contradiction is already present in the stated Local-to-Global Embedding Principle. For every hereditarily symmetric surjection $f\twoheadrightarrow B$, that theorem purports to construct a function $s\to A$ satisfying $f\circ s=\operatorname{id}_B$. Thus it asserts that every surjection has a right inverse, which is equivalent over $\mathsf{ZF}$ to the Axiom of Choice and is stronger than the injection required by the Partition Principle. It therefore cannot prove $\mathsf{PP}+\neg\mathsf{AC}$. The supporting localization and gluing arguments fail independently: a condition cannot be strengthened while deleting coordinates outside a fixed support; the fixed support is replaced in the proof by element-dependent supports; the proposed coherent antichains cannot exist below arbitrary Boolean values; minimality in an arbitrary ground-model well-order does not imply equivariance; right inverses need not agree on overlaps; and countably many finite supports need not combine into a finite support. The route-unification theorem and the proposed witness to $\neg\mathsf{AC}$ have further defects. This replacement retains only standard background facts about symmetric extensions and records the obstructions. It asserts no model of $\mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$ and no consistency or independence result.

math.LO↗

arXiv2026

Arithmetical Complexity and Absoluteness of Rigidity Phenomena for Ulam Sequences

We analyse the arithmetical complexity and forcing absoluteness of natural statements about Ulam sequences. For positive integers $a<b$, membership in $U(a,b)$ and the increasing enumeration of $U(a,b)$ are uniformly primitive recursive. We encode the finite interval-with-periodic-mask descriptions used in rigidity results and formalise the finite-window rigidity theorem for the family $U(1,n)$: for every prescribed linear window, one finite collection of residue-class-dependent pattern data works for all sufficiently large parameters $n$. This family statement has a $Π^0_3$ upper bound in the arithmetical hierarchy; for a fixed window its complexity is $Σ^0_2$. For an individual $U(a,b)$, eventual periodicity of the gap sequence (with positive period) and finiteness of specified residue classes are $Σ^0_2$, while rational upper-density, lower-density, and exact-density assertions have $Π^0_3$ upper bounds. These are classifications by upper bounds, not completeness claims. Since the resulting sentences are arithmetical, their truth is unchanged by set forcing. This semantic forcing invariance is distinguished from proof-theoretic conservativity over stronger set theories. Finally, if the gaps of $U(a,b)$ are eventually periodic with positive period, then $U(a,b)$ is Presburger-definable; hence $(\mathbb{N},+,\mathrm{U}_{a,b})$ is decidable, NIP, dp-minimal, and does not interpret full arithmetic.

math.LO↗

arXiv2026

A countable-support symmetric iteration separating PP from AC

We study a countable-support Cohen symmetric seed model and a proposed symmetric-iteration approach to separating the Partition Principle $\mathsf{PP}$ from the Axiom of Choice. The previously claimed final construction is withdrawn. Its package forcing targets right inverses, and therefore a localized splitting principle stronger than the ordinary localized Partition Principle. For the fixed seed parameter $S=A^ω$, that splitting principle together with $\mathsf{SVC}(S)$ implies $\mathsf{AC}$, contrary to the intended preservation of a non-well-orderable Cohen set. Independently, the final non-well-orderability argument confuses stabilization of forcing names with an action inside one fixed generic extension, and the limit-stage $\mathsf{SVC}(S)$ argument relies on an invalid truncation lemma. The retained positive result is the Cohen symmetric seed [ \mathcal{N}\models\mathsf{ZF}+\mathsf{DC}+\mathsf{SVC}(S)+\neg\mathsf{AC}, \qquad S=(A^ω)^{\mathcal{N}}. ] The package and iteration sections are preserved only as a record of the superseded approach. No model of $\mathsf{ZF}+\mathsf{DC}+\mathsf{PP}+\mathsf{AC}_{\mathsf{WO}}+\neg\mathsf{AC}$ is claimed here.

math.LO↗

arXiv2026

Limit Filters and Dependent Choice in Countable-Support Symmetric Iterations

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 $ω$, countable-intersection closure is imposed in the definition. The resulting limit filters are normal and $ω_1$-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.

math.LO↗

arXiv2026

Explicit separation of quadratic irrationals from the middle-third Cantor set

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 $1$ in a quadratic irrational, and consequently a quantitative separation from the middle-third Cantor set. Those conclusions are withdrawn. The argument used a decomposition defined only after finite exit had been assumed in order to prove finite exit, and the proposed quadratic-field-uniform Thue--Mahler estimate did not follow from the cited $S$-unit theory: the relevant set $S$ varies with the minimal polynomial, and the displayed $S$-unit quotient does not retain the power-of-$3$ exponent without an additional normalization. This replacement records the defects and retains the unconditional part of the argument. For a quadratic irrational whose orbit exits, we prove the exact exit-to-distance identity and an elementary height-dependent clearance bound. We also give a corrected universal bound for a single $L$-run and show that an exceptionally long $R$-to-$L$ transition produces a bounded value of a binary quadratic norm form at a power of $3$. A finite-prefix decomposition identifies the additional uniform orbit-counting theorem that would be required for a genuine separation result. No assertion that a quadratic irrational lies outside the Cantor set is made here.

math.NT↗

arXiv2026

The Modal Logic of Finitely Symmetry-Preserving Iterated Extensions is Exactly S4

We determine the ZF-provable modal logic of the modality $\Box_{\mathrm{sym}}$, where $\Box_{\mathrm{sym}}φ$ means '$φ$ holds in every finite symmetry-preserving iteration' of the symmetric method. We prove that the exact logic is S4. Soundness (axioms T and 4) follows from reflexivity and transitivity of the underlying accessibility relation. Exactness is obtained by (i) a non-amalgamation lemma showing that axiom (.2) fails for finite symmetry-preserving iterations (no common finite symmetry-preserving iteration above the parent), and (ii) a $p$-morphism/finite-frame realization producing, within ZF, models whose $\Box_{\mathrm{sym}}$-theory matches any finite reflexive-transitive frame.

math.LO↗

arXiv2026

A One-Step Cascade Symmetric Model: Rank-$1$ Packets, Binary Shielding, and the Even Exact-Cardinality Profile

We introduce a one-step cascade symmetric system whose local symmetry geometry is organized by finite $ρ$-closed windows and one-step stars rather than by rowwise-independent toggles. The resulting symmetric model isolates a new $ZF + DC + \neg \mathrm{BPI}$ geometry in which rank-$1$ hereditarily symmetric reals admit a packet normalization theorem over countable $ρ$-closed supports. The technical center of the paper is the finite star-span lemma and the associated rank-$1$ packet calculus. From this we obtain a normalization theorem and a two-layer coding consequence for rank-$1$ reals (in the metatheory, via a well-orderable base of packets). We then apply the same binary fresh-support shielding pattern to prove $\neg C_2$, hence $\neg AC_{\mathrm{fin}}$, and therefore the failure of every even $C_n$ (where $C_n$ denotes the principle that every family of nonempty $n$-element sets admits a choice function). On the odd side, the present bounded packet calculus remains dyadic: support-fixed local actions factor through finite $2$-groups, bounded support-equivariant quotients of finite local orbits have power-of-two size, and trace-separated bounded rigid ternary families admit canonical selectors within a fixed finite trace window. Accordingly, the odd exact-cardinality profile remains open beyond the current local binary machinery.

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