The Implicit Definition of the Set-Concept

Abstract

Once Hilbert asserted that the axioms of a theory `define` theprimitive concepts of its language `implicitly'. Thus whensomeone inquires about the meaning of the set-concept, thestandard response reads that axiomatic set-theory defines itimplicitly and that is the end of it. But can we explainthis assertion in a manner that meets minimum standards ofphilosophical scrutiny? Is Jané (2001) wrong when hesays that implicit definability is ``an obscure notion''? Doesan explanation of it presuppose any particular view on meaning?Is it not a scandal of the philosophy of mathematics that no answersto these questions are around? We submit affirmative answers to allquestions. We argue that a Wittgensteinian conception of meaninglooms large beneath Hilbert's conception of implicit definability.Within the specific framework of Horwich's recent Wittgensteiniantheory of meaning called semantic deflationism, we explain anexplicit conception of implicit definability, and then go on toargue that, indeed, set-theory, defines the set-conceptimplicitly according to this conception. We also defend Horwich'sconception against a recent objection from the Neo-Fregeans Hale and Wright (2001). Further, we employ the philosophicalresources gathered to dissolve all traditional worries about thecoherence of the set-concept, raisedby Frege, Russell and Max Black, and whichrecently have been defended vigorously by Hallett (1984) in hismagisterial monograph Cantorian set-theory and limitationof size. Until this day, scandalously, these worries havebeen ignored too by philosophers of mathematics.