2 papers
math.LO2026
Existence of bases implies the axiom of choice, a foundation-free proof
Gabriel Fernandes, Renan Maneli Mezabarba, Vinicius de Oliveira Rodrigues
We prove that, in Zermelo--Fraenkel set theory with the axiom of Foundation removed, the statement that every vector space has a basis implies the Axiom of Choice, concluding that…
math.LO2026
A Borel Concept Class of VC Dimension One with a Non-PAC Consistent Learner in ZFC
Mateus Jesus de Arruda Campos, Gabriel Fernandes, Vinicius de Oliveira Rodrigues
The fundamental theorem of statistical learning states that, under suitable measurability assumptions, finite Vapnik--Chervonenkis (VC) dimension guarantees that every proper consi…