k-trace definability finds universal k-NIP theories inside NIP bases
Hilbert space, Urysohn space and generic hypergraphs appear as the canonical objects k-trace definable in real closed fields or vector space
math.LO
Logic
Logic, set theory, point-set topology, formal mathematics
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Natural numbers object is projective in the free topos
Categorical proof recovers Makkai's argument establishing countable choice for intuitionistic higher-order logic
· “Makkai's lost proof of projectivity of N in the free topos”
SOP2 equals SOP3, closing a 2004 question
SOP1, SOP2, and SOP3 are one property, collapsing the bottom of the finite-cycle hierarchy.
Every Ackermannian Goodstein walk ends at zero
Any legal way of rewriting a number still terminates, thanks to one base-change-maximal notation.
Nonexistence of 3-ladders at ℵ₂ matches Mahlo consistency
Ditor's 1984 question on whether the size bound ℵ_{n-1} is attained for n=3 is independent of ZFC.
Proof validity becomes a real number in this logic
A family of calculi converges to linear logic as p grows, giving neural networks a sound quantitative logic.
Modal group theory under homomorphisms equals true arithmetic
For finitely presented groups the Gödel numbers of true modal sentences match those of true arithmetic up to computable mapping.
Group embeddings validate modal logic S4.2
Modal group theory shows propositional validities under embeddings are exactly S4.2 and interprets true arithmetic.
Hoare logic equivalent to second-order logic with first-order predicates
Partial correctness assertions for iterative programs hold in one system exactly when they hold in the other.
Two-sorted ultrametrics form Fraïssé class with universal limit
Distance-carrying embeddings on points and ordered distances produce a homogeneous space U universal for all separable ultrametrics.
Finitely generated G outside C ties amenability to Thompson F
The group is amenable exactly when F is, yet its minimal line actions lack a Borel transversal for conjugacy.
Localic Esakia duality becomes fully constructive
Heyting frames meet Esakia locales through conic frames, with no prime ideal theorem.
The Turing jump gets a fixed point—as an ideal
Uniform limit oracles miss the fixed ideal and restart the jump hierarchy.
· “Jump Closure and Limit Uniformization in the Ideal Completion of the Turing Degrees”
Weak set theory still proves transitive closures exist
It makes the no-infinity version equivalent to KP and settles two open questions.
Stone duality is computable for quasi-Polish spaces
Overt discrete and compact Hausdorff spaces encode each other with computable functors and adjunctions
· “Effective quasi-Polish categories of overt discrete spaces and compact Hausdorff spaces”
One modal valuation principle is exactly the omega-jump axiom
Extending assignments to valuations, Geach axioms, GL, and Barcan definability all collapse to ACA0^+.
Very slow-growing Borel graphs are hyperfinite
A growth exponent near 0.1523 settles the hyperfiniteness question for amenable actions of that growth.
Congruence preservation equals Boolean-algebra closure
Stable preorders match lattice closure; on N, Z, and p-adics, recognizable sets make the test finite.
A 13-bit word proves exact and unique automaton complexity differ
Exact complexity is 6, unique is 7 - the first length where the two measures disagree.
· “Exact versus unique nondeterministic automatic complexity”
Beyond ℵ1, triangle-free graph forcing is no longer Cohen
A new theorem pins down when generic structures built from Fraïssé classes are Cohen, with a sharp boundary at ℵ1.
Random-generic equivalence is never essentially free
Even a positive-measure piece collapses to one class under any Borel map into an essentially free relation.
Regular bisimulations don't cover consistent anti-foundation axioms
New graph equivalences with the set property yield consistent axioms that refute every bisimulation-based AFA.
Well-orderable reals coincide with all five smallness ideals
A Baire-category axiom unifies the main smallness ideals and gives perfect subsets of positive-measure sets.
· “Ideals, Well-Orderable Families, and Baire Category in Truss's Feferman-Type Model”
Categories with computability bases model dependent type theory
Presheaves are the type families; their categories of elements supply the Sigma-objects of a dependent-type model.
Index registers are pinned down by the program path alone
For BSS RAMs over first-order structures, index contents depend on program, input length, and path, not on the structure.
· “Abstract computation over first-order structures. Extras: From programs to decision trees I”
Closed two-letter words are derivable with stricter grammar rules
The result sharpens earlier proofs that O_2 is a multiple context-free language.
Rigid real closed fields exist through transcendence degree ℵ1
Height bounds on curves pick independent coordinates, producing fields with no automorphisms for every degree 2–ℵ1.
STRP lifts from finite-index subgroups to overgroups
The generic Cantor-action property climbs finite-index inclusions; its Borel rank lies between the second and fourth arithmetical levels.
· “Finite-Index Lifting of Strong Topological Rokhlin Property and Descriptive Complexity”
Two commuting maps with bounded fibers give hyperfinite relations
New proof: any two commuting bounded-to-one Borel maps generate an equivalence relation that is hyperfinite.
· “Hyperfiniteness of bounded-to-one actions of commutative monoids”
One rule decides every totality question in grounded arithmetic
OGA gives every totality question a certified yes-or-no status and reclassifies its Gödel sentence as a safe, priced fiction.
Every definable endomorphism of ordered abelian groups is rational
Pure-language proof answers Kourovka Notebook Problem 18.16; parameter-free case included, expansion languages can fail.
· “A note on definable endomorphisms of ordered abelian groups”
Full-strength arithmetic defines its own truth predicate
Machine-checked proofs close the square: what RGA proves, it affirms true and provable from within.
Large feedforward nets: output becomes one fixed number
The output concentrates around a fixed $\psi$ with error probability decaying like $e^{-cn}$; only aggregate weight statistics are needed.
· “A concentration result for multilayer feedforward neural networks”
Opposite doctrines turn quotients into comprehensions
The paper proves the swap is exact: each doctrine has quotients exactly when its opposite has comprehensions.
· “Quotients-comprehensions duality in relational doctrines”
Three labels classify every ω1-anchored model
A trichotomy and a stationary dichotomy unite two analyses of minimal Vaught counterexamples.
Hyperlow reals are exactly those that make Δ¹₁ relativize uniformly
If true, hyperlowness fully determines when every Δ¹₁(f) set is an f-section of a single Δ¹₁ relation.
Full-modal formulas get complete logic-programming answers
Arbitrary modal formulas as programs and queries; computed answers equal declarative correctness.
Hausdorff limits stay definable over non-Archimedean fields
For bounded Lipschitz-cell families, the proof uses dense completions and tame pairs.
· “Definability of Hausdorff Limits for Lipschitz Cells in O-minimal Structures”
Tree decompositions decide hyperfiniteness on hyperbolic boundaries
If a hyperbolic graph splits as a tree of hyperfinite bags, its boundary action is hyperfinite too.
· “Hyperfiniteness of boundary actions via tree decompositions”
Eliminating hyperimaginaries is exactly Barr-exactness
Type-definable equivalence relations collapse to definable ones exactly when the pro-completion is Barr-exact.
Comeager hereditary families of compact sets are big in ZFC
Matheron and Zelený's problem is settled without definability restrictions or extra set-theoretic hypotheses.
Martin's Axiom freezes the ω₁² Ramsey failure against c.c.c
Once a witness to ω₁²↛(ω₁²,3)² exists, every c.c.c. forcing preserves it in models of MA.
Division by 3 stays unprovable in weak arithmetic with standard truths
Even with all universal arithmetic truths added, some element has no remainder modulo 3.
Stability in S-act classes is exactly the LO monoid property
Superstability adds weak noetherianity; ideal generating sizes give the full stability spectrum.
· “On the abstract elementary class of acts with pure embeddings”
Amalgamation is decidable for semantic Horn classes
A 2ExpTime algorithm decides the amalgamation property for finite models of universal Horn sentences, even with unbounded relation arity.
· “Deciding Amalgamation Beyond Arity Two: The Semantic Horn Case”
All equivalence relations on FIN[∞]₂ become canonical
Every map on a front of the Gowers space reduces to a classified canonical type, with an exact count.
Modal Kleene algebras are exactly twist structures over contact lattices
A two-way translation and a topological duality extend the twist-product representation to modal operators.
· “A Modal Expansion of Kleene Algebras via Twist Structures”
Measure truth as the weight of satisfied predicates
Bundle-and-weight model yields rigorous fuzzy membership, modal necessity, and a sorites solution.
Near-homomorphisms of quantum logics are exact on Boolean blocks
Blockwise stability: approximate joins and meets yield genuine homomorphisms on each block, though gluing can fail
Dense unit-circle trajectories add no new open definable sets
A conditional proof shows the torus curves are too sparse to change the real field's topology.
Loeb-equivalence question resolved for all internal probability spaces
Two Loeb-equivalent internal spaces always share an internal probability measure on their generated algebra.
· “Loeb Equivalence for General Internal Probability Spaces”
Every full n-branching ordinal tree has one of four decidable theories
Height collapses to four canonical models, each with effective quantifier elimination.
· “The elementary theory of full n-branching ordinal trees with successor functions”
No infinite compact T makes Spec(C(T)) an Esakia space
A diagonal counterexample plus a grid theorem closes the open question for all infinite compact Hausdorff spaces.
The Q-point count can never be ℵ₀ or ℵ₁
New proof pins the number of Q-point ultrafilters to finite, 2^𝔡, or 2^𝔠—and ties infinite cases to Q-measures and Tukey-top ultrafilters.
Gaussian-style elimination solves convex-set inequalities
Smallest convex solutions reduce to finite formulas; a broad class of lamination hulls becomes computable.
· “Solving polynomial inequalities over spaces of convex sets and applications”
Super tree property on any countable segment of singulars
A forcing model makes the super tree property hold at the first θ successors of singulars, for any countable θ.
· “Strong Tree Properties Along Many Segments of Successors of Singulars”
Relative property (T) forces a Bauer–Poulsen dichotomy
Transitive groups: invariant-measure simplices are Bauer or Poulsen, settling the exchangeability question.
· “Relative Property (T), simplices of invariant measures, and existentially closed models”
Adding quantifiers makes conditional logic unaxiomatizable
The logic of set-valued selection functions encodes arithmetic, so no recursive calculus can be complete.
· “Stalnaker's logical problem of conditionals is unsolvable”
A syntactic proof puts S4 inside the logic BBI
A cancelling reverse translation makes the embedding faithful and parametric across BBI extensions.
· “Embedding Modal Logics into Logics of Bunched Implications”
Logic J gets strong completeness over topological models
Strong completeness for polymodal provability logic, with a transfer to GLP over the same spaces.
Adding a reflection scheme to WKL0 proves no new Π₁₁ theorems
Every countable model of the base theory extends to WKL0 plus full reflection; no Π₁₁ theorem is lost
· “L\'evy-Montague reflection is Pi¹₁-conservative over mathsf{WKL}₀”
Contact relations match closed relations on Stone spaces
The paper proves one-to-one correspondence for lattices and join-semilattices, extending the Boolean case.
· “Contact relations on bounded distributive lattices and related structures”
Prediction errors define a new degree structure finer than truth-table
The degree structure decomposes into a truth-table spine and a downward-closed random region, with random reals never derandomized.
· “Sequential-Innovation Reducibility and the Innovation Spectrum”
Choice-free duality: pOMLs are dual to upper Vietoris c-spaces
Every partial orthomodular lattice is represented by a spectral space of filters and vice versa—no Axiom of Choice needed.
All cohesive powers of one computable graph compute the double jump
A companion linear order has cohesive powers with no computable presentation, extending a classical limitation of arithmetic.
SPL embeds into S
A corrected translation sending p to p∧□p matches each propositional calculus to its modal calculus sequent by sequent.
· “Embeddings of Propositional Logics into the Provability Logics S and D”
Convolution turns tame Banach lattices into wild structures
Infinite abelian subgroups force the tree property of the second kind, and non-discrete groups are unstable.
Trace reflection lifts weak counting to full asymptotic classes
For polynomial-graph incidence structures, the denominator k is provably minimal and K_{k,n} is forbidden.
16 unknowns keep Hilbert's Tenth Problem undecidable over any quadratic field
The paper reduces the unsolvable 10-variable integer problem to 16-variable equations over integer rings, and to 15 over real quadratic fiel
· “On Diophantine equations over the integer rings of quadratic fields”
One theorem unifies Lawson and de Groot dualities
A single formula swaps continuity for completeness and stability for cocompleteness.
S4 De Morgan algebras are dually equivalent to pairwise Stone spaces
Recovers each algebra from biclopen sets of its prime-filter spectrum; yields completeness for modal FDE.
Under a Church thesis, second-order quantifiers collapse
In ACA*_0 + ACT, formula classes sharpen and a reflection separation follows without choice.
· “Normal Forms and Uniform Reflection under the Arithmetical Church Thesis”
Any choice of positional bases still produces the real numbers
A digit-string construction makes double names like 0.5 = 0.4999... a theorem, not a convention.
This paper proves that the class of all order-theoretic trees is well-quasi-ordered by…
Every infinite sequence of order-theoretic trees contains two trees such that one maps homomorphically into the other, with a proof that…
Twelve logical separations via one oracle-topos bridge
Total Weihrauch reducibility is exactly the Lawvere-Tierney modality order, so each computability separation yields a new constructive axiom
· “Modified realizability subtoposes and total Weihrauch reducibility”
δ-closure is decided by abstract Hardy-field germs
A new curve selection theorem turns infinitely many jet conditions into one germ-level direction for differential topology.
Under CH, dense-set choice splits Katětov classes
Two countable dense sets in an uncountable product can yield non-equivalent coordinate ideals, so D is not a harmless choice.
· “Dense-set dependence in the Katv{e}tov order for uncountable coordinate ideals”
Bounded-to-1 commuting functions generate hyperfinite Borel graphs
The connectedness relation of such a graph is a countable union of finite Borel equivalence relations.
Simple groups of finite Morley rank with a large torus are Chevalley
A weaker K-group assumption lets the earlier proof transfer, giving the final step for generic simple groups.
· “The c{S}ic{s} Kebap Theorem: A Generic Identification Theorem for Groups of Finite Morley Rank”
Uniformly stationary Kaplansky classes are stable
The uniform Kaplansky property puts flat Mittag-Leffler modules and Drinfeld bundles in the stable range.