Monomial ideals and independence of $\mathrm{I}\Sigma_2$

TL;DR

This paper establishes a connection between a simplified version of Maclagan's theorem on monomial ideals and the logical strength of the system IΣ₂, revealing a phase transition threshold and emphasizing the theorem's combinatorial aspects.

Contribution

It introduces a miniaturized version of Maclagan's theorem and proves its equivalence to the 1-conservativity of IΣ₂, providing new insights into the theorem's combinatorial nature.

Findings

Miniaturized Maclagan's theorem is equivalent to 1-Con(IΣ₂).

Identifies a phase transition threshold for the theorem.

Highlights the combinatorial aspects of Maclagan's theorem.

Abstract

We show that a miniaturised version of Maclagan's theorem on monomial ideals is equivalent to $\mathrm{1}{-}\mathrm{Con}(\mathrm{I}\Sigma_2)$ and classify a phase transition threshold for this theorem. This work highlights the combinatorial nature of Maclagan's theorem.