Infinite lotteries, spinners, and the applicability of hyperreals

TL;DR

This paper defends the use of hyperreal probabilities in infinite lotteries by addressing criticisms, showing how to avoid arbitrariness, and clarifying the distinction between internal and external hyperreal measures.

Contribution

It demonstrates that hyperreal probabilities can be justified using specific models and clarifies the nature of measures, countering objections based on arbitrariness and underdetermination.

Findings

Hyperreal probabilities can be modeled without arbitrariness using Kanovei-Shelah or saturated models.

External hyperreal measures are parasitic and lead to underdetermination.

Internal hyperfinite measures are well-defined and not underdetermined.

Abstract

We analyze recent criticisms of the use of hyperreal probabilities as expressed by Pruss, Easwaran, Parker, and Williamson. We show that the alleged arbitrariness of hyperreal fields can be avoided by working in the Kanovei-Shelah model or in saturated models. We argue that some of the objections to hyperreal probabilities arise from hidden biases that favor Archimedean models. We discuss the advantage of the hyperreals over transferless fields with infinitesimals. In the second part we will analyze two underdetermination theorems by Pruss and show that they hinge upon parasitic external hyperreal-valued measures, whereas internal hyperfinite measures are not underdetermined.