Model Theory of Ultrafinitism I: Fuzzy Initial Segments of Arithmetics

TL;DR

This paper develops a model theory for ultrafinitism using fuzzy initial segments of natural numbers, introducing a proof theory and semantics to handle feasibly consistent theories within a finite framework.

Contribution

It proposes a novel model of ultrafinitistic arithmetic based on fuzzy initial segments, along with a proof theory and semantics for feasibly consistent theories.

Findings

Introduces fuzzy initial segments as models of ultrafinitistic arithmetics

Develops a proof theory compatible with ultrafinitism

Provides a semantics for feasibly consistent theories

Abstract

This article is the first of an intended series of works on the model theory of Ultrafinitism. It is roughly divided into two parts. The first one addresses some of the issues related to ultrafinitistic programs, as well as some of the core ideas proposed thus far. The second part of the paper presents a model of ultrafinitistic arithmetics based on the notion of fuzzy initial segments of the standard natural numbers series. We also introduce a proof theory and a semantics for ultrafinitism through which feasibly consistent theories can be treated on the same footing as their classically consistent counterparts. We conclude with a brief sketch of a foundational program, that aims at reproducing the transfinite within the finite realm.