Vopěnka's Alternative Set Theory as a framework for feasible numbers

Abstract

Vopěnka's Alternative Set Theory has been considered as a framework for modelling vague notions. This paper takes feasibility, pertaining to numbers as per some of Yessenin-Volpin's work, and tries to assess how this notion could be modelled in the Alternative Set Theory. The route explored in detail consists in an attempt to model Dummett's weakly infinite, weakly finite totalities, the coherence of which Dummett takes to be decisive of the coherence of Yessenin-Volpin's foundational conception in general. The outcome of the particular analysis turns out to be negative: the Alternative Set Theory could be taken as a model of some, but not quite all features of Yessenin-Volpin's feasible numbers. Given that only one particular approach is explored here, the outcome has limited bearing on the more general questions of coherence of ultrafinitist position, or indeed the possibilities of modelling feasible numbers.