The hall of mirrors
The hall of mirrors

Summary/Abstract: This paper examines the concept of set, drawing a sharp distinction between the mathematical and logical set concepts. The mathematical concept, which underpins ZFC and related systems, is based on the iterative conception of sets formed in stages. By contrast, the logical concept, originating in Frege’s work, identifies sets with the extensions of concepts. While the naive logical concept leads to paradoxes, two prominent attempts to salvage it are Russell’s Theory of Types (TT) and Quine’s New Foundations (NF). Through an analysis of Specker’s equiconsistency result, we argue that NF, despite its aim of ontological parsimony, implicitly relies on a typed structure. Evaluating NF as a Quinean explication, we conclude that it fails on pragmatic grounds due to its mathematical limitations, particularly the failure of the Axiom of Choice. Finally, we suggest that Russell’s type theory, by virtue of its structural coherence and its deep connection to the λ-calculus, offers a superior analysis of the logical concept of set.

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