Impredicativity and Paradox

Abstract

Michael Dummett famously asked how the serpent of inconsistency entered Frege's paradise. He himself blamed the impredicative nature of second-order quantification, while many others focused on the inflationary nature of the axiom. Axiom V is, after all, the denial of a higher-order generalization of Cantor's theorem. Predicativists do not deny this, but they block the derivation of the relevant generalization in predicative fragments of second-order logic. Unfortunately, there is more than one higher-order generalization of Cantor's theorem, and one of them remains a theorem in predicative fragments of higher-order logic. Our recommendation to predicativists is to respond that only one of them supports the cardinality gloss we associate with Cantor's theorem and that it is, in fact, false. The other remains a theorem of predicative fragments of higher-order logic but its derivability seems more closely related to the Grelling's paradox than to cardinality considerations.

Ground and Grain

“…This, however, is not the case. As discussed recently by Uzquiano (2019), certain types of Cantorian arguments can be carried out using only predicative plural comprehension, and such arguments turn out to be possible in the present case as well. The matter is again most easily illustrated with the variant using arbitrary conjunction presented in section 2.3.…”

“…The case of arbitrary conjunction is obtained by letting φ ( p , q ) be p ≺ q , and ψ ( pp ) be ⋀ pp . (The use of a binary relation in ( S ′ φ , ψ ) is modeled on a version of Cantor’s theorem in Bernays (1942); see Uzquiano (2019, p. 212) for further discussion related to predicative comprehension. )…”

Operands and Instances

“…This additional strength of I I Qv can be harnessed to show an inconsistency result which requires only predicative instances of PC. Instead of the Russell-Myhill theorem, the relevant derivation makes use of the following result, adapted from [14]: Proposition 33. (p, ϕ(pp)) ↔ p ≺ pp can be shown to be inconsistent using one instance of plural comprehension, namely the one for condition ¬ (q, q).…”