戈德尔不完备定理董世平大学应用数学系.ppt

戈德爾不完備定理董世平中原大學應用數學系 二十世紀數理邏輯之重大成果 二十世紀數理邏輯之重大成果 * 上世紀末19011903190419301931 Georg Cantor(1845-1919)David Hilbert(1862-1943)Berfrand Russell(1872-1970)Ernst Zermelo(1871-1956)Kurt G?del(1907-1978)Kurt G?del : (1)有理數可數 (N0) (2)實數不可數 (C ) (3)無窮數無窮 (4)超越數不可數: 23個問題(第二屆世界數學家會議) #1. 連續假設 C = N1  #2. 算數公設之一致性(Hilbert Program) #10. 不定方程式之演算法: 羅素詭論 : 選擇公設 : 述詞邏輯完備 : 不完備定理 193619391963196619701971 Alan Turing(1912-1954)Alonzo Church(1903-1995)Kurt G?del Paul J. CohenAbraham Robinson( ? -1974)Juri MatijasevichStephen Cook : 涂林機(全備計算機): 邱池論(定義計算): 連續假設與選擇公設之一致性: 連續假設與選擇公設之獨立性: 非標準分析: 不定方程式不可決定: NP-完備性 P=NP? On January 14, 1978, Kurt G?del died in Princeton in his seventy-first year.There are those who believe that he was the most brilliant mind of the twentieth century. When Harvard University gave him an honorary degree*, the citation described him as “discoverer of the most significant mathematical truth of this century, incomprehensible to laymen, revolutionary for philosophers and logicians.” * 1952 It is with this analysis, and its impact on the minds of such men as John von Neumann and others, that the theoretical concepts and the analysis of the digital computer in the modern sense began. It remains true to this very day that the theoretical description of what can be computed in general and its more penetrating analysis are rooted in the soil of mathematical logic which G?del turned over for the first time in his memoir of 1931. The great abstract logical work of G?del had a striking outcome. In analyzing the forma; machinery of G?del’s description of what could be obtained by step-by-step procedures, the brilliant young English logician Alan Turing identified the results of such procedures-the general recursive functions-with the outcomes of what could be computed on a machine in general. 第一不完備定理　　　任何一個足夠強的一致公設系統，必定是不完備的。　　　即除非這個系統很簡單（所以能敘