• P. Geach  “Class and concept”, in: Philosophical Review, vol. 64, 1955


  • John L. Bell  “Frege's Theorem in a Constructive Setting”, in: Journal of Symbolic Logic, vol. 64, no. 2, 1999

By Frege's Theorem is meant the result, implicit in Frege's Grundlagen, that, for any set E, if there exists a map ν from the power set of E to E satisfying the condition
∀XY[ ν(X) = ν(Y) ⇔ X ≈ Y] ,
then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map ν be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., will involve no use of the law of excluded middle.


  • 井川俊彦  『数学基本用語小事典』、日本評論社、2006年



姉妹編『研究社−ロングマン イディオム英和辞典』は持っていたが、こちらはまだだったので入手。今回入手した辞典の元となったLongman Dictinary of Phrasal Verbsを持っていたので入手が遅れたものと思われる。

  • F. A. Muller  “The Implicit Definition of the Set-Concept”, in: Synthese, vol. 138, no. 3, 2004

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 ofphilosophical 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 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 his magisterial monograph Cantorian set-theory and limitation of size. Until this day, scandalously, these worries have been ignored too by philosophers of mathematics.

Humeの原理を理解する上で、implicit definitionに対する理解を深めておくことは、大切なようだ。