The supertask of an infinite lottery

TL;DR

This paper models Hansen's supertask involving an infinite sequence of gods selecting a number, revealing the supertask's underdetermined nature and the limitations of finitely additive probability measures in this context.

Contribution

It provides a mathematical analysis of Hansen's supertask, demonstrating its underdetermined scenarios and the weak notion of uniformity for finitely additive measures.

Findings

The supertask admits many consistent scenarios.

Uniformity for finitely additive measures is unreasonably weak.

The supertask's structure leads to underdetermined probability assignments.

Abstract

We mathematically model the supertask, introduced by Hansen, in which an infinity of gods together select a random natural number by each randomly removing a finite number of balls from an urn, leaving one final ball. We show that this supertask is highly underdetermined, i.e. there are many scenarios consistent with the supertask. In particular we show that the notion of uniformity for finitely additive probability measures on the natural numbers emerging from this supertask is unreasonably weak.