Infinite Lotteries, Spinners, Applicability of Hyperreals†

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 Paper II we analyze two underdetermination theorems by Pruss and show that they hinge upon parasitic external hyperreal-valued measures, whereas internal hyperfinite measures are not underdetermined.

Saturated models of first-order many-valued logics

This paper is devoted to the problem of existence of saturated models for first-order many-valued logics. We consider a general notion of type as pairs of sets of formulas in one free variable that express properties that an element of a model should, respectively, satisfy and falsify. By means of an elementary chains construction, we prove that each model can be elementarily extended to a $\kappa $-saturated model, i.e. a model where as many types as possible are realized. In order to prove this theorem we obtain, as by-products, some results on tableaux (understood as pairs of sets of formulas) and their consistency and satisfiability and a generalization of the Tarski–Vaught theorem on unions of elementary chains. Finally, we provide a structural characterization of $\kappa $-saturation in terms of the completion of a diagram representing a certain configuration of models and mappings.

On the uniqueness of cellular injectives

A. Avilés and C. Brech proved an intriguing result about the existence and uniqueness of certain injective Boolean algebras or Banach spaces. Their result refines the standard existence and uniqueness of saturated models. They express a wish to obtain a unified approach in the context of category theory. We provide this in the framework of weak factorisation systems. Our basic tool is the fat small object argument.