From geometry to geology: An invitation to mathematical pluralism through the phenomenon of independence

TL;DR

This paper advocates for mathematical pluralism in set theory by examining independence phenomena, forcing, and set-theoretic geology, highlighting the diversity of models and universes that challenge the notion of a single ultimate set universe.

Contribution

It introduces a framework connecting independence in set theory with mathematical pluralism, using forcing and geology to explore multiple set-theoretic universes and their properties.

Findings

Independence phenomena lead to a plurality of models.

Forcing creates a multiverse of diverse set-theoretic universes.

Set-theoretic geology studies local neighborhoods of the multiverse.

Abstract

This paper explores how a pluralist view can arise in a natural way out of the day-to-day practice of modern set theory. By contrast, the widely accepted orthodox view is that there is an ultimate universe of sets $V$, and it is in this universe that mathematics takes place. From this view, the purpose of set theory is "learning the truth about $V$." It has become apparent, however, that the phenomenon of independence - those questions left unresolved by the axioms - holds a central place in the investigation. This paper introduces the notion of independence, explores the primary tool ("soundness") for establishing independence results, and shows how a plurality of models arises through the investigation of this phenomenon. Building on a familiar example from Euclidean geometry, a template for independence proofs is established. Applying this template in the domain of set theory leads to a consideration of forcing, the tool par excellence for constructing universes of sets. Fifty years of forcing has resulted in a profusion of universes exhibiting a wide variety of characteristics - a multiverse of set theories. Direct study of this multiverse presents technical challenges due to its second-order nature. Nonetheless, there are certain nice "local neighborhoods" of the multiverse that are amenable to first-order analysis, and \emph{set-theoretic geology} studies just such a neighborhood, the collection of grounds of a given universe $V$ of set theory. I will explore some of the properties of this collection, touching on major concepts, open questions, and recent developments.