Universism and Extensions of V

TL;DR

This paper proposes a novel interpretative method for Universists to understand extension talk in set theory, challenging the view that only multiverses can account for such discourse and offering new philosophical insights.

Contribution

It introduces a new interpretation of extension talk for Universists, connecting it to $V$-logic, outer models, and impredicative class theories, thus enriching the philosophical debate.

Findings

Provides a method to interpret extension talk within a Universist framework.

Shows how the interpretation relates to satisfaction in outer models.

Argues the technique is philosophically virtuous and opens new discussion avenues.

Abstract

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by providing a hitherto unexamined way for a Universist to interpret talk that seems to necessitate the addition of sets to $V$. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk ($V$-logic), and show how it captures satisfaction in `ideal' outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist.