# 连续统假设

$\aleph_0 < |S| < 2^{\aleph_0}.$ 假設選擇公理是對的，那就會有一個最小的基數$\aleph_1$大於$\aleph_0$，而連續統假設也就等價於以下的等式：

$2^{\aleph_0} = \aleph_1.$

## 作為希爾伯特第一問題

1900年，大衛·希爾伯特以「連續統假設是否成立」作為「希爾伯特第一問題」。Kurt Godel和Paul Cohen確定了連續統假設在ZFC系統下，加上了選擇公理，也不能證明或證否。 连续统假设简记CH。选择公理简记AC。

## 廣義連續統假設

CHGCH都獨立於ZFC，不過Sierpiński證明了ZF+GCH可以推導出選擇公理，換句話說，不存在ZF+GCH但AC不成立的公設系統。

$A < B \to 2^A \leqslant 2^B$.

$A < B \to 2^A < 2^B \!$

GCH意味着这个严格的不等式对无限序数和有限序数都成立。

## 參考資料

• Cohen, P. J. Set Theory and the Continuum Hypothesis. W. A. Benjamin. 1966.
• Cohen, Paul J. The Independence of the Continuum Hypothesis. Proceedings of the National Academy of Sciences of the United States of America. Dec. 15, 1963, 50 (6): 1143–1148.
• Cohen, Paul J. The Independence of the Continuum Hypothesis, II. Proceedings of the National Academy of Sciences of the United States of America. Jan. 15, 1964, 51 (1): 105–110.
• Dales, H. G.; W. H. Woodin. An Introduction to Independence for Analysts. Cambridge. 1987.
• Foreman, Matt. Has the Continuum Hypothesis been Settled? (PDF). 2003 [February 25, 2006].
• Freiling, Chris. Axioms of Symmetry: Throwing Darts at the Real Number Line. Journal of Symbolic Logic. 1986, 51 (1): 190–200.
• Gödel, K. The Consistency of the Continuum-Hypothesis. Princeton University Press. 1940.
• Gödel, K.: What is Cantor's Continuum Problem?, reprinted in Benacerraf and Putnam's collection Philosophy of Mathematics, 2nd ed., Cambridge University Press, 1983. An outline of Gödel's arguments against CH.
• Maddy, Penelope. Believing the Axioms, I. Journal of Symbolic Logic. June 1988, 53 (2): 481–511.
• Martin, D. (1976). "Hilbert's first problem: the continuum hypothesis," in Mathematical Developments Arising from Hilbert's Problems, Proceedings of Symposia in Pure Mathematics XXVIII, F. Browder, editor. American Mathematical Society, 1976, pp. 81–92. ISBN 0-8218-1428-1
• McGough, Nancy. The Continuum Hypothesis.
• Woodin, W. Hugh. The Continuum Hypothesis, Part I. Notices of the AMS. 2001a, 48 (6): 567–576.
• Woodin, W. Hugh. The Continuum Hypothesis, Part II. Notices of the AMS. 2001b, 48 (7): 681–690.