Adding Up to Nothing: Coarse Reasoning and the Vanishing St. Petersburg Paradox
TL;DR for operators The paper is not a magic trick that turns an infinite expected value into a finite one. The ordinary St. Petersburg expectation still diverges. Anyone claiming otherwise has either missed the point or found a very ambitious way to lose a philosophy seminar. What the paper actually does is more interesting. Takashi Izumo defines a coarse-grained version of arithmetic in which numbers are first mapped into finite “grains,” each grain is represented by a selected internal value, and addition is performed through repeated projection to those representatives.1 Under this operation, an increment can become too small to move the current coarse state. That phenomenon is called absorption. Repeated absorption produces inertness: further additions keep arriving, but the represented total stops changing. ...