The Implicit Definition of the Set-Concept

Once Hilbert asserted that the axioms of a theory ‘define‘ the primitive concepts of its language ‘implicitly’. Thus when someone inquires about the meaning of the set-concept, the standard response reads that axiomatic set-theory defines it implicitly and that is the end of it. But can we explain this assertion in a manner that meets minimum standards of philosophical scrutiny? Is Jané (2001) wrong when he says that implicit definability is “an obscure notion”? Does an explanation of it presuppose any particular view on meaning? Is it not a scandal of the philosophy of mathematics that no answers to these questions are around? We submit affirmative answers to all questions. We argue that a Wittgensteinian conception of meaning looms large beneath Hilbert’s conception of implicit definability. Within the specific framework of Horwich’s recent Wittgensteinian theory of meaning called semantic deflationism, we explain an explicit conception of implicit definability, and then go on to argue that, indeed, set-theory, defines the set-concept implicitly according to this conception. We also defend Horwich’s conception against a recent objection from the Neo-Fregeans Hale and Wright (2001). Further, we employ the philosophical resources gathered to dissolve all traditional worries about the coherence of the set-concept, raised by Frege, Russell and Max Black, and which recently have been defended vigorously by Hallett (1984) in his magisterial monograph Cantorian set-theory and limitation of size. Until this day, scandalously, these worries have been ignored too by philosophers of mathematics.