The article provides a significant contribution to the ongoing debate on “how the serpent (of inconsistency) entered Frege’s paradise”, i.e., the system presented in Frege’s Grundgesetze. Two competing explanations have traditionally been proposed, each of which identifies an axiom as the origin of the contradiction, in virtue of a syntactic feature and a (supposedly semantic) gloss. On the one hand (e.g., G. Boolos, Whence the Contradiction?, Aristotelian Society Supplementary Volume 67 , (1993), pp. 213–233.), the Cantorian view attributed the inconsistency to Basic Law Vb

(BLVb) $$ \begin{align} \forall X \forall Y (\epsilon X = \epsilon Y \to \forall x (Xx \leftrightarrow Yx)) \end{align} $$

as imposing an injective function from the second-order domain to the first-order one: this request negates an instance of an higher-order formalisation of the “Not one-one” version of Cantor’s theorem

(C) $$ \begin{align} \lnot \exists \iota \forall X \forall Y (\iota X = \iota Y \to \forall x (Xx \leftrightarrow Yx)), \end{align} $$

which prevents such a function and suggests an interpretation of the contradiction in terms of the incompatible cardinality requests on the domains. On the other hand (e.g., M. Dummett, Frege: Philosophy of mathematics. Harvard University Press, (1991).), the predicativist view attributed the inconsistency to the second-order comprehension axiom scheme ( $\exists X \forall x (Xx \leftrightarrow \phi (x))$ ) due to its impredicativity, glossed in terms of vicious circularity.

The Cantorian perspective apparently benefits from the availability of a “Not onto” higher-order formalisation of Cantor’s theorem

(B) $$ \begin{align} \lnot \exists R \forall X \exists x \forall y (Rxy \leftrightarrow Xy), \end{align} $$

which states that no binary relation can encode a surjection of a domain onto its power set. In the Cantorian perspective, B is merely a reformulation of C, as suggested by their equivalence modulo (full) Comprehension and Choice. Moreover, while C is derivable, both directly and from B, only via impredicative comprehension, B can also be derived in (different versions of) predicative Second-order logic (see R. G. Heck, The consistency of predicative fragments of Frege’s Grundgesetze der Arithmetik. History and Philosophy of Logic , vol. 17 (1996), no. (1–2), pp. 209–220, K. F. Wehmeier, Consistent fragments of Grundgesetze and the existence of non-logical objects. Synthese , vol. 121 (1999), no. 3, pp. 309–328, and F. Ferreira and K. F. Wehmeier, On the Consistency of the Δ11-CA Fragment of Frege’s Grundgesetze. Journal of Philosophical Logic , vol. 31 (2002), no. 4, pp. 301–311.), and seems to play a role in the derivation of a lesser-known Russell-like contradiction. The article argues that these results do not necessarily support the Cantorian perspective, suggesting an anti-Cantorian reading for them.

In short, B can be predicatively derived by reductio ad absurdum of the claim that every monadic concept X has an object x as a proxy, namely

(R) $$ \begin{align} \forall X \exists x \forall y (Rxy \leftrightarrow Xy). \end{align} $$

Absent impredicative comprehension, R is compatible with the Fregean assumption that every concept has an extension, since we cannot specify a relation $\eta $ between concepts and objects that behaves as required in G. Boolos, The consistency of Frege’s Foundations of Arithmetic, in J. J. Thomson, On Being and Saying: Essays for Richard Cartwright, (1987), pp. 3–20. above: $\forall X \exists ! x \forall Y (Y \eta x \leftrightarrow \forall y (Xy \leftrightarrow Yy))$ . From a Cantorian perspective, the inconsistency of R depends on it being the negation of an instance of B and can be explained in the same way as the inconsistency of BLVb, as the negation of an instance of Cantor’s theorem. The predicativist could still argue that, if we wanted to derive R from the $\eta $ -version of BLV, we would require an instance of impredicative (dyadic) comprehension. Furthermore, in a predicativist setting, B is consistent with BLV—i.e., with the negation of C—and, therefore, it supports neither the gloss of Cantor’s theorem in terms of the cardinality of the domains, nor the corresponding Cantorian explanation of the inconsistency of R.

To identify an alternative explanation of this inconsistency, the author suggests to focus on the derivation of R from a first-order scheme:

$$ \begin{align*} \exists x \forall y (Rxy \leftrightarrow \phi(y)), \end{align*} $$

from which a first-order version of Russell’s paradox also follows—by instantiating $\phi (y)$ with $\lnot Ryy$ . From a non-Cantorian perspective, the contradictory character of this scheme depends on the unavailability of a satisfaction predicate for the first-order language—as is evident in the derivation of Grelling’s paradox from one of its instances (which only requires a bijection between the open first-order formulas and a countable number of constants).

Incidentally, similar results and an analogous dialectic could also arise if we replaced second-order logic with plural logic. In general, with respect to the original debate, the article’s results not only highlight the Cantorian difficulty in explaining any version of the paradox not involving C, but also seem to support the main predicativist concern left open in the previous debate—on the role of the cardinality gloss as the source of the inconsistency. Due to the availability of general models of SOL, the cardinality consequences of C only characterise the proper subset of its standard models. Therefore, it is questionable whether they should be included in the interpretation of a theorem, or in the explanation of a negative syntactic result such as a contradiction. As a confirmation of such a model-theoretic remark, it has been shown that Russell’s paradox and C—whose derivations mainly coincide—can be proven from a version of BLV that is restricted to definable (and therefore countable) concepts (cf. A. C. Paseau, Did Frege commit a cardinal sin?. Analysis , 75 (2015), no. 3, pp. 379–386.).

On the contrary, the analogy with Grelling’s paradox sheds new light on the relation between Russell’s paradox and the limitative results on the definability of semantic notions (see V. Halbach, Axiomatic theories of truth. Cambridge University Press, 2014 as well as V. Halbach and S. Zhang, Yablo Without Gödel. Analysis , vol. 77 (2017), no. 1, pp. 53–59.). A proof of Tarski’s theorem has been recently provided within a “Fregean-inspired ZF” (in J. D. Hamkins, Fregean abstraction in Zermelo-Fraenkel set theory: a deflationary account. (2022), arXiv preprint arXiv:2209.07845.), which uses the derivation of Russell’s paradox, and does not require a full coding of the syntax, but only a device—such as Fregean abstraction—to represent formulas by objects. Therefore, due to its generality, such a non-Cantorian explanation could aspire to uniformly replace the original predicativist one.