Continuity in Potential Infinite Models

TL;DR

This paper introduces a model of simple type theory with potential infinite carrier sets where functions are inherently continuous without relying on topological or domain theoretic concepts, using a novel factor system approach.

Contribution

It presents a new model based on factor systems that generalize direct and inverse systems, capturing potential infinity without actual infinite sets or topology.

Findings

Functions are automatically continuous in the model

Factor systems enable construction of limits within the expansion process

The model aligns with the concept of potential infinity

Abstract

We introduce a model of simple type theory with potential infinite carrier sets. The functions in this model are automatically continuous, as defined in this paper. This notion of continuity does not rely on topological concepts, including domain theoretic concepts, which essentially use actual infinite sets. The model is based on the concept of a factor system, which generalizes direct and inverse systems. A factor system is basically an extensible set indexed over a directed set of stages, together with a predecessor relation between object states at different stages. The function space, when considered as a factor system, expands simultaneously with its elements. On the one hand, the space is subdivided more and more, on the other hand, the elements increase and are defined more and more precisely. In addition, a factor system allows the construction of limits that are part of its expansion process and not outside of it. At these limits, elements are indefinitely large or small, which is a contextual notion and a substitute for elements that are infinitely large or small (points). This dynamic and contextual view is consistent with an understanding of infinity as a potential infinite.