Higher-Order Platonism and Multiversism

TL;DR

This paper explores higher-order Platonism within the multiverse view of set theory, using Zalta and Linsky's Object Theory to formalize and address philosophical criticisms of multiversism.

Contribution

It formalizes higher-order Platonism in the context of Hamkins' multiversism using Object Theory, aiding in addressing key philosophical criticisms.

Findings

Object Theory faithfully expresses Hamkins' multiverse conception

Embedding clarifies the articulation and skeptical issues of multiversism

Supports the relationship between multiversism and set-theoretic practice

Abstract

Joel Hamkins has described his multiverse position as being one of `higher-order realism -- Platonism about universes', whereby one takes models of set theory to be actually existing objects (vis-\`a-vis `first-order realism', which takes only sets to be actually existing objects). My goal in this paper is to make sense of the view in the very context of Hamkins' own multiversism. To this end, I will explain what may be considered the central features of higher-order platonism, and then will focus on Zalta and Linsky's Object Theory, which, I will argue, is able to faithfully express Hamkins' conception. I will then show how the embedding of higher-order platonism into Object Theory may help the Hamkinsian multiversist to respond to salient criticisms of the multiverse conception, especially those relating to its articulation, skeptical attitude, and relationship with set-theoretic practice.