Internality, transfer, and infinitesimal modeling of infinite processes

TL;DR

This paper clarifies the distinction between internal and external hyperreal measures in probability theory, demonstrating that internal hyperfinite measures are well-defined and comparing their advantages over other ordered fields.

Contribution

It proves that internal hyperfinite measures are not underdetermined and evaluates the expressive power of transferless ordered fields versus hyperreals.

Findings

Internal hyperfinite measures are not underdetermined.

Robinson's transfer principle applies only to internal entities.

Hyperreal probabilities are more expressive than those over transferless ordered fields.

Abstract

A probability model is underdetermined when there is no rational reason to assign a particular infinitesimal value as the probability of single events. Pruss claims that hyperreal probabilities are underdetermined. The claim is based upon external hyperreal-valued measures. We show that internal hyperfinite measures are not underdetermined. The importance of internality stems from the fact that Robinson's transfer principle only applies to internal entities. We also evaluate the claim that transferless ordered fields (surreals, Levi-Civita field, Laurent series) may have advantages over hyperreals in probabilistic modeling. We show that probabilities developed over such fields are less expressive than hyperreal probabilities.