集合论国际【力迫法】 (2-1)

论文短，影响深远，得国际大奖，这三点同时满足的很多人估计只能想到纳什的博弈论论文，但实际上1963年Paul Cohen发明力迫法获得菲尔兹奖的工作只有6页，登在PNAS上面，应该是菲尔兹奖级别的最短的论文。

THE INDEPENDENCEOF THE CONTINUUM HYPOTHESIS

BY PAUL HEN*

DEPARTMENT OF MATHEMATICS, STANFORD UNIVERSITY

Communicαled by Kurl Gōdel, September 30, 1968

This is the first of two notes in which we outline a proof of the fact that the Con-tinuum Hypothesis cannot be derived from the other axioms of set theory, including the Axiom of Choice. Since Gödel³ has shown that the Continuum Hypothesis is consistent with these axioms，the independence of the hypothesis is thus estab-lished. We shall work with the usual axioms for Zermelo-Fraenkel set theory，² and by Z-F we shall denote these axioms without the Axiom of Choice，(but with the Axiom of Regularity). By a model for Z-F we shall always mean a collection of actual sets with the usual ∊relation satisfying Z-F. We use the standard defini-tions' for the set of integers ω， ordinal, and cardinal numbers.

THEOREM 1. There αre models for Z-F in ωhich the folloωing occur：

(1) There is α set α， α ⊆ ω such thαt α is not constructible in the sense of reference S，yet the A xiom of Choice αnd the Generαlized Conlinum Hypothesis both hold.

(2) The continuum (i.e.，℘(ω) ωhere ℘ meαns poωer set) hαs no ωell-ordering.

(3) The Axiom of Choice holds，bul ℵ ≠ 2ℵ⁰.

(4) The Axiom of Choice for countαble pαirs of elemenls in ℘(℘(ω))fαils.

就这样一篇论文，构造了一个满足ZFC但是其中连续统假设为假的模型（6页的这篇给出了力迫模型的构造，下一篇也是6页，证明力迫模型的确满足我们想要的的条件），和哥德尔差不多三十年前的结果共同解决了希尔伯特第一问题，直接让Cohen获得了菲尔兹奖

数学联邦政治世界观提示您：看后求收藏（同人小说网http://tongren.me），接着再看更方便。