Vop\v{e}nka's Alternative Set Theory as a framework for feasible numbers

TL;DR

This paper investigates whether Vopěnka's Alternative Set Theory can model feasible numbers as defined by Yessenin-Volpin, focusing on Dummett's weakly infinite and finite totalities, and finds limited success in capturing all features.

Contribution

It explores the application of Alternative Set Theory to model feasible numbers, specifically analyzing Dummett's weakly infinite and finite totalities, revealing partial modeling capabilities.

Findings

Partial modeling of feasible numbers achieved

Alternative Set Theory cannot fully capture all features of feasible numbers

Limited scope of approach suggests further research needed

Abstract

Vop\v{e}nka'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.