Transfer of canonical hyperreal evaluations to improved number systems

Prove that in any improved or alternative theory of summation and integration built on a different infinite number system (such as the surreal numbers), the canonical evaluations produced by hyperreal summation and integration transfer; specifically, establish that the improper integral of 1/x from 1 to infinity equals log(+), where + denotes the simple, canonical infinite element of that alternative number system.

Background

The paper proposes extending the hyperreal-based theory of summation and integration to other infinite number systems, noting that versions based on the surreals may be promising. The author suggests that canonical answers obtained in the hyperreal framework should persist in any improved formulations.

This conjecture aims to ensure stability of key evaluations, such as improper integrals, across different foundational number systems so that results like ∫_1^∞ 1/x dx receiving a canonical infinite-logarithmic value remain invariant under such changes.