| 释义 |
Cantor, n.2
(ˈkæntɔː(r) ‖ ˈkantor)
Math. The name of Georg Cantor (1845–1918), Russian-born German mathematician, used in the possessive and attrib. to designate various concepts relating to the theory of sets and infinite numbers arising out of his work, as Cantor('s) (ternary) set, the set of points left by removing from a line of unit length all points whose distance from one end is greater than 1/3 and less than 2/3, then removing similarly the middle third of the two segments so formed, and so on indefinitely.
1902Proc. London Math. Soc. XXXIV. 286 H. J. S. Smith's ternary derived set is to all intents and purposes the same as Cantor's ternary set of numbers. Ibid., The generalization of Cantor's set. Ibid. XXXV. 248 Cantor's Theorem. Every set of intervals on a straight line is countable, provided no two overlap. 1903B. Russell Princ. Math. xlii. 347 The thesis of the present chapter is, that Cantor's continuum is free from contradictions. Ibid. 527 There is a one-one relation of all ranges of propositions to some propositions, which is directly contradictory to Cantor's theorem. 1906Proc. London Math. Soc. 2nd. Ser. IV. 272 We can prove..that every ordinal number is a Cantor's ordinal number. 1940Amer. Math. Monthly XLVII. 549 (heading) Distances between points of the Cantor set. 1953A. A. Fraenkel Abstr. Set Theory ii. 94 The highly comprehensive answer, often called Cantor's theorem, runs:..To any set S there exist sets having larger cardinals than S; in particular, the set US, whose elements are all the subsets of S, is of a larger cardinal than S. 1960P. R. Halmos Naive Set Theory xxv. 101 The contradiction, based on the assumption that there is such a set [of all cardinal numbers], is known as Cantor's paradox. 1975I. Stewart Concepts Mod. Math. ix. 143 Prior to Cantor's theorem, mathematicians had become accustomed to thinking of transcendental numbers as being very rare, because they seldom seemed to use any. 1979D. R. Hofstadter Gödel, Escher, Bach (1980) v. 142 When α is irrational, the bands shrink to points, of which there are infinitely many, very sparsely distributed in a so-called ‘Cantor set’—another recursively defined entity which springs up in topology. 1987Nature 24 Dec. 695/1 The resulting function resembles a mathematical function called a Cantor function or, more picturesquely, a ‘devil's staircase’.
Hence Canˈtorian a., of or pertaining to Cantor or his work; having the attributes of a Cantor set; also Canˈtorean a.
1912Whitehead & Russell Principia Mathematica II. v. 614 A compact Dedekindian series is said to possess ‘Dedekindian continuity’; such series have many important properties. They are a wider class than series possessing Cantorian continuity. 1946Mind LV. 367 His theory made it impossible to prove the existence of the continuum and thus curtailed the Cantorean theory of sets. 1963W. V. Quine Set Theory 294 Classes that are not Cantorian behave unconventionally. 1982W. S. Hatcher Logical Found. Math. vii. 225 A set x is ‘Cantorian’ if it is similar to its set of unit subsets. |