Consistent ultrafinitist logic

TL;DR

This paper explores ultrafinitist logic, proposing a proof system that restricts reasoning to short objects, aiming to resolve paradoxes and inconsistencies associated with infinitary reasoning.

Contribution

It introduces a preliminary proof system based on Curry-Howard isomorphism tailored for ultrafinitist logic and analyzes how classical theorems change under these constraints.

Findings

Certain classical theorems no longer hold in ultrafinitist systems

Opposite statements become provable within the system

Some paradoxes are explained as consequences of transfinite reasoning

Abstract

Ultrafinitism postulates that we can only compute on relatively short objects, and numbers beyond certain value are not available. This approach would also forbid many forms of infinitary reasoning and allow to remove certain paradoxes stemming from enumeration theorems. However, philosophers still disagree of whether such a finitist logic would be consistent. We present preliminary work on a proof system based on Curry-Howard isomorphism. We also try to present some well-known theorems that stop being true in such systems, whereas opposite statements become provable. This approach presents certain impossibility results as logical paradoxes stemming from a profligate use of transfinite reasoning.