设 Conway's Game of Life H H 23 0 A 1 John Horton Conway
Conway’s Game of Life John Horton Conway
设雾 Turing Machine ON COMPUTABLE NUMBERS.WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM By A.M.TURING. [Received 28 May,1936.-Read 12 November,1936.] The "computable"numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable,computahle
230 A. M. TUKING [Nov. 12, ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM By A. M. TURING. [Received 28 May, 1936.—Read 12 November, 1936.] The "computable" numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers. it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable, computable predicates, and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique. I hope shortly to give an account of the relations of the computable numbers, functions, and so forth to one another. This will include a development of the theory of functions of a real variable expressed in terms of computable numbers. According to my definition, a number is computable if its decimal can be written down by a machine. In §§ 9, 10 I give some arguments with the intention of showing that the computable numbers include all numbers which could naturally be regarded as computable. In particular, I show that certain large classes of numbers are computable. They include, for instance, the real parts of all algebraic numbers, the real parts of the zeros of the Bessel functions, the numbers IT, e, etc. The computable numbers do not, however, include all definable numbers, and an example is given of a definable number which is not computable. Although the class of computable numbers is so great, and in many Avays similar to the class of real numbers, it is nevertheless enumerable. In § 81 examine certain arguments which would seem to prove the contrary. By the correct application of one of these arguments, conclusions are reached which are superficially similar to those of Gbdelf. These results f Godel, " Uber formal unentscheidbare Satze der Principia Mathematica und ver- •vvandter Systeme, I" . Monatsheftc Math. Phys., 38 (1931), 173-198. Turing Machine
设 ENIAC 第一台通用电子计算机ENIAC(1947年)
ENIAC 第⼀台通⽤电⼦计算机 ENIAC（1947年）
设 David Hilbert (1862-1943 “希尔伯特问题”（1900) 23个数学难题： 第2问题： 为算数建立完备的公理体系 第10问题： 给出解整数方程的算法 希尔伯特
David Hilbert (1862-1943) ! “希尔伯特问题”（1900） ! 23个数学难题： ! 第2问题： 为算数建⽴完备的公理体系 ! 第10问题： 给出解整数⽅程的算法 希尔伯特
设雾 Euclid (300 BC) THE ELEMENTS 第一界 厂成已角形之已三角切甲乙 形之在形内及切在形外者先以面 此卷将自切形在园之内外及作同在 形内切 麻形居他直狼形内而此形之各角切献 消养 十 鼎 欧几里德
Euclid (300 BC) 欧几里德