Paradoxes of infinity and beyond: sets, quantification, and cantor's domain principle

Harding, Tarryn

Paradoxes of infinity and beyond: sets, quantification, and cantor's domain principle

Thesis / Dissertation

2026

Publisher

Unversity of Cape Town

Department

Department of Philosophy

Faculty

Faculty of Humanities

Abstract

A central tenet of Cantor's set theory is what Michael Hallett (1986) calls the Domain Principle: “each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite.” One way to read the Principle is as asserting a necessary condition for successful quantification over the elements of infinite totalities – namely, that these form sets. However, if true, this would entail that we can never simultaneously quantify over all sets (and thus, a fortiori, over everything at once): for the assumption that we do succeed in quantifying over all sets, when paired with the truth of the Domain Principle, entails the existence of a universal set, and thus Russell's Paradox. The question of whether the set-theoretic paradoxes give us reasons to endorse Non Absolutism about quantifiers – where this is the claim that our ordinary, first-order quantifiers fail to range over absolutely everything – forms the core of my dissertation, which I explore by way of detailed expositions of two possible responses to the paradoxes. According to the first, the origins of the contradictions lies with a naïve principle of Collapse, according to which any plurality forms a set. Typically, opponents of Collapse also endorse quantifier Absolutism – the claim that it is possible for our quantifiers to range over an absolutely comprehensive domain – which commitment, together with their rejection of Collapse, entails that the absolutists in question also reject the Domain Principle. However, I contend that these theorists face the Objection from Vagueness, whereby the lack of a principled distinction between sets and non-sets threatens to imbue the boundaries of the former domains with a species of what Michael Dummett dubs ‘haziness'; such haziness, in turn, undermines determinate quantification over any ostensibly comprehensive domain of sets, by rendering it unclear what the instances of the resulting quantified statements are. The second response to the paradoxes involves rehabilitating Collapse by means of appropriately interpreted modal operators (the idea being, roughly, that any plurality of ‘actual' elements forms a ‘potential' set); the question of how to accommodate the ‘new' sets secured by Collapse is, in turn, resolved according to whether these are absorbed in expanding predicate extensions – a view which, by virtue of retaining absolutely general quantification, is known as Third-Way Absolutism – or, whether it is the range of our quantifiers that expand (or, alternatively, de-restrict) in order to accommodate the ‘new' sets in question. The latter form of Restrictionist Non-Absolutism – according to which our quantifiers are systematically restricted, relative to a domain comprising absolutely everything – is vulnerable to Tim Williamson's Objection from Semantic Theorizing: the complaint is that the restrictionist, in attempting to give a semantics for her quantifiers, ends up falling back, illicitly, on absolutely general thought. However, I argue that Expansionist Non-Absolutism is vulnerable to a similar set of objections: firstly, according to the Objection from Implicit Actualism, the well-behaved – albeit limited – domains ranged over by the expansionist's quantifiers naturally lend themselves to a reconstruction of an absolutely general domain (a similar objection – the Objection from Super-Meanings – can be levelled against the third-way absolutist: the complaint is that the series of well-behaved predicate expansions accompanying Collapse prompts the question of what prevents us from merging each of these extensions into an over-arching ‘super extension'). Secondly, according to the Objection from Crypto-Restrictionism, the logical behaviour of the expansionist's quantifiers lends itself less to their being understood as unrestricted quantifiers ranging over domains that are subsequently subject to expansion, and more to their being understood as restricted, relative to a domain comprising absolutely everything. At the heart of these latter two objections is the idea that the non-absolutist fails to distinguish adequately the sort of generality expressed by her modal operators – which generalize across or between quantifier domains – and that expressed by her quantifiers, which range over the elements of a single, limited domain. In my conclusion, I discuss one way we might cash out this distinction – namely, in terms of instance- versus non-instance-based generality – and argue that an instance-based account of quantificational generality lends support to the validity of the Domain Principle, especially in the modal context. More generally, though, I contend that giving an account of the distinction between modal and quantificational generality is essential to the success of the modal theories under consideration – and that, so far, no such fully-worked out account is forthcoming.

Keywords

Philosophy

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