Higher-Order Platonism and Multiversism

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

Forcing and the Universe of Sets: Must we lose insight?

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 forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems to necessitate the addition of subsets 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 analyse extant interpretations of such talk, and argue that while tradeoffs in naturality have to be made, they are not too severe

Are Large Cardinal Axioms Restrictive?

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper, I argue that whether or not large cardinal axioms count as maximality principles depends on prior commitments concerning the richness of the subset forming operation. In particular I argue that there is a conception of maximality through absoluteness, on which large cardinal axioms are restrictive. I argue, however, that large cardinals are still important axioms of set theory and can play many of their usual foundational roles

Indeterminateness and `The' Universe of Sets: Multiversism, Potentialism, and Pluralism

In this article, I survey some philosophical attitudes to talk concerning `the' universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views

Forcing and the Universe of Sets: Must we lose insight?

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 forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems to necessitate the addition of subsets 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 analyse extant interpretations of such talk, and argue that while tradeoffs in naturality have to be made, they are not too severe

Forcing and the Universe of Sets: Must we lose insight?

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 forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems to necessitate the addition of subsets 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 analyse extant interpretations of such talk, and argue that while tradeoffs in naturality have to be made, they are not too severe

Intrinsic Justification for Large Cardinals and Structural Reflection

We deal with the complex issue of whether large cardinals are intrinsically justified principles of set theory (we call this the Intrinsicness Issue). In order to do this, we review, in a systematic fashion, (1.) the abstract principles that have been formulated to motivate them, as well as (2.) their mathematical expressions, and assess the justifiability of both on the grounds of the (iterative) concept of set. A parallel, but closely linked, issue is whether there exist mathematical principles able to yield all known large cardinals (we call this the Universality Issue), and we also test principles for their responses to this issue. Finally, we discuss the first author's Structural Reflection Principles (SRPs), and their response to Intrinsicness and Universality. We conclude the paper with some considerations on the global justifiability of SRPs, and on alternative construals of the concept of set also potentially able to intrinsically justify large cardinals

Universism and extensions of V

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 developing a 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