by contradictions. (H. Poincaré [Les mathématiques et la logique III, Rev. métaphys. morale 14 (1906) p. 316]) When the objects of discussion are linguistic entities [...] then that collection of entities may vary as a result of discussion about them. A consequence of this is that the "natural numbers" of today are not the same as the "natural numbers" of yesterday. (D. Isles ) There are at least two different ways of looking at the numbers: as a completed infinity and as an incomplete infinity... regarding the numbers as an incomplete infinity offers a viable and interesting alternative to regarding the numbers as a completed infinity, one that leads to great simplifications in some areas of mathematics and that has strong connections with problems of computational complexity. (E. Nelson ) During the renaissance, particularly with Bruno, actual infinity transfers from God to the world. The finite world models of contemporary science clearly show how this power of the idea of actual infinity has ceased with classical (modern) physics. Under this aspect, the inclusion of actual infinity into mathematics, which explicitly started with G. Cantor only towards the end of the last century, seems displeasing. Within the intellectual overall picture of our century ... actual infinity brings about an impression of anachronism. (P. Lorenzen) See also Limit (mathematics) Cardinality of the continuum References Sources "Infinity" at The MacTutor History of Mathematics archive, treating the history of the notion of infinity, including the problem of actual infinity. Aristotle, Physics Bernard Bolzano, 1851, Paradoxien des Unendlichen, Reclam, Leipzig. Bernard Bolzano 1837, Wissenschaftslehre, Sulzbach. Georg Cantor in E. Zermelo (ed.) 1966, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Olms, Hildesheim. Richard Dedekind in 1960 Was sind und was sollen die Zahlen?, Vieweg, Braunschweig. Adolf Abraham Fraenkel 1923, Einleitung in die Mengenlehre, Springer, Berlin. Adolf Abraham Fraenkel, Y. Bar-Hillel, A. Levy 1984, Foundations of Set Theory, 2nd edn., North Holland, Amsterdam New York. Stephen C. Kleene 1952 (1971 edition, 10th printing), Introduction to Metamathematics, North-Holland Publishing Company, Amsterdam New York. . H. Meschkowski 1981, Georg Cantor: Leben, Werk und Wirkung (2. Aufl.), BI, Mannheim. H. Meschkowski, W. Nilson (Hrsg.) 1991, Georg Cantor – Briefe, Springer, Berlin. Abraham Robinson 1979, Selected Papers, Vol. 2, W.A.J. Luxemburg, S. Koerner (Hrsg.), North Holland, Amsterdam. Infinity Concepts in metaphysics Philosophy of mathematics