Against Cumulative Type Theory

Abstract

Standard Type Theory, STT, tells us that ( ) is well-formed iff = + 1. However, Linnebo and Rayo (2012) have advocated for the use of Cumulative Type Theory, CTT, which has more relaxed type-restrictions: according to CTT, ( ) is well-formed iff > . In this paper, we set ourselves against CTT. We begin our case by arguing against Linnebo and Rayo's claim that CTT sheds new philosophical light on set theory. We then argue that, while CTT's type-restrictions are unjustifiable, the type-restrictions imposed by STT are justified by a Fregean semantics. What is more, this Fregean semantics provides us with a principled way to resist Linnebo and Rayo's Semantic Argument for CTT. We end by examining an alternative approach to cumulative types due to Florio and Jones (2021); we argue that their theory is best seen as a misleadingly formulated version of STT.10 This extends Degen and Johannsen's (2000: §4.1) results concerning Z. Linnebo and Rayo (2012: 289, n.28) cover only Z without Foundation. The bound + 2 < is needed as abbreviates ∃ +1 (∀ +2 ( +2 ( +1 ) ↔ +2 ( )) ∧ +1 ( )).11 Linnebo and Rayo (2012: 284, 289) mention differences ( 1) and ( 2) themselves, but they do not mention (3). 65 Zr is equivalent to Potter's (2004) theory Z; this is strictly stronger than Zermelo's Z. 66 Notation: we let ' ⊆ ∈ ℎ' abbreviate '( ⊆ ∧ ∈ ℎ)'; similarly for other infix predicates. 67 See Button (forthcoming: §3) for proofs. 68 Compare these with Degen and Johannsen (2000: 149 Ext, 153 Nullity).

Two conceptions of absolute generality

“…Here we followFlorio & Jones 2021, 50-51. 9 This view is in keeping withButton & Trueman 2021, Williamson 2003, and Williamson 2013…”

“…We call these relational answers to the Analytical Question. The present investigation of relational answers was spurred by the critical discussion in Button & Trueman 2021, to which we return below (Sect. 4).…”

“…In standard systems of cumulative type theory, one natural thought is that domains of any types can be compared for inclusivity (Krämer 2017, Button & Trueman 2021. These comparisons are enabled by the cross-type identity relations (Degen & Johannsen 2000, Linnebo & Rayo 2012, which cumulative type theory arguably provides: 10 This argument for the possibility of absolute generality in the sense of maximally strong generality in cumulative type theory presupposes R=U.…”

“…This depends in turn on what forms of quantification make sense. For example, some cumulative systems include quantifiers that, to put it intuitively, check all entities in the domain that are below a certain type (Degen & Johannsen 2000, Linnebo & Rayo 2012, Button & Trueman 2021. By appropriate choice of type, these quantifiers will allow exhaustion of any domain and absolute generality will therefore be possible.…”

“…Formally, ∀x i (x(a) ↔ x(b)) where i is the first type above the types of both a and b. These definitions are employed inDegen & Johannsen 2000, Linnebo & Rayo 2012, Krämer 2017, and Button & Trueman 2021…”