2022/02/02 by James A. Walsh, Walsh, James · 1 citation
Computer Science · Mathematics · #Advanced Topology and Set Theory #Complexity and Algorithms in Graphs #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2202.01174
openalex publication_date 2022/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let's fix a reasonable subsystem T of arithmetic; why are natural extensions of T pre-well-ordered by consistency strength? In previous work, an approach to this question was proposed. The goal of this work was to classify the recursive functions that are monotone with respect to the Lindenabum algebra of T. According to an optimistic conjecture, roughly, every such function must be equivalent to an iterate ConTα of the consistency operator in the limit. In previous work the author established the first case of this optimistic conjecture; roughly, every recursive monotone function is either as weak as the identity operator in the limit or as strong as ConT in the limit. Yet in this note we prove that this optimistic conjecture fails already at the next step; there are recursive monotone functions that are neither as weak as ConT in the limit nor as strong as ConT2 in the limit. In fact, for every α, we produce a function that is cofinally as strong as ConαT yet cofinally as weak as ConT.