Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.32078

Oracle-Tree Forcing and Wilfs Inequality for Thirteen Left Elements

Michel Gaspar
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study filter-Laver forcing on omegaxomega with successor sets containing final quadrants. The observation retaining the minimum coordinate and orientation is a complete projection. Its finite nontrivial hidden-label quotients are Cohen, whereas the full offset quotient has Boolean density equal to the ground-model dominating number dV . For the associated grid-meager ideal I_box, we prove cov(I_box) = add(M) for the usual meager ideal M, and obtain further cardinal bounds. We also analyze threshold games, oracle degrees and the distinction between positive and filter-large certificate sets: recurrent filter-large acceptance admits refinements whose every branch succeeds. As an arithmetic application, we prove Wilfs inequality c \leq 13e for numerical semigroups with thirteen elements below the conductor c, where eis the embedding dimension. A complete arithmetic proof, including a bounded exact computation, is given in the appendix. Its terminating certificate procedures supply a computable quadrant tree and uniform refinements for every forcing condition. Finite support kernels describe persistence of the registered proofs; arithmetic absoluteness explains precisely why forcing organizes these certificates without replacing their arithmetic justification.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michel Gaspar. 2026-09-25. Oracle-Tree Forcing and Wilfs Inequality for Thirteen Left Elements. https://arxiv.org/abs/2609.32078

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Freely adding one layer of quantifiers to a Boolean doctrine

We describe the layer of quantifier alternation depth at most one of the quantifier completion of a Boolean doctrine over a small category. This amounts to a doctrinal version of Herbrand's theorem for formulas with quantifier alternation depth at most one modulo a universal theory. The resulting construction satisfies a universal property that makes it the free QA-one-step Boolean doctrine. To achieve this version of Herbrand's theorem, we characterize, within the doctrinal setting, the classes $A$ of quantifier-free formulas for which there is a model $M$ such that $A$ is precisely the class of formulas whose universal closure is valid in $M$.

math.LO↗
arXiv2026

Generalized Borel Sets

Generalizing classical descriptive set theory opens foundational questions about the Borel hierarchy. In this paper we systematically study those questions, working in the general framework of Polish-like spaces relative to an uncountable cardinal $κ$, possibly singular, satisfying $2^{<κ}=κ$. We provide fundamental properties of the $κ^+$-Borel hierarchy of any regular Hausdorff space of weight at most $κ$, and establish sufficient conditions for its non-collapse. We highlight a unique phenomenon that arises in the case of singular cardinals, namely, the existence of a second, distinct Borel hierarchy, the $κ$-Borel hierarchy: we prove that it is strictly finer than the $κ^+$-Borel hierarchy, and then characterize the precise relationship between the two. Finally, for regular cardinals, we resolve three questions about the behavior of the $κ^+$-Borel hierarchy on subspaces of the generalized Baire space ${}^κκ$, constructing various models via forcing where several nontrivial constellations for the length of the $κ^+$-Borel hierarchy on the space are realized.

math.LO↗
arXiv2026

The complexity of being monitorable

We study monitorable sets from a topological standpoint. In particular, we use descriptive set theory to describe the complexity of the family of monitorable sets in a countable space $X$. When $X$ is second countable, we observe that the family of monitorable sets is $Π^0_3$ and determine the exact complexities it can have. In contrast, we show that if $X$ is not second countable then the family of monitorable sets can be much more complex, giving an example where it is $ Π^1_1$-complete.

math.LO↗