Consistency of the Continuum Hypothesis
Consistency of the Continuum Hypothesis
复制标题
连续体假说的一致性
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
Bruce W. Rogers
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文献类型：
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作者：
Bruce W. Rogers
集合理论的基本结果之一是，自然数的功率集的基数与实数的基数相同，这实际上大于自然的基础。 ：对于任何集合X，X的基数严格少于X的功率集的基数。由于有一个无限的集合，例如Naturals，功率集的每个应用都给出了每个无限套件的更大数字。 x我们分配了一个基数אx。 α<βIFF的α<α<ββ的一种方法是，我们不知道哪些红衣主教在秩序中进行。 2א0，但我们不知道康托尔之间是否有任何枢机主教。该陈述被称为连续假设[CH]，因为一个人可以证明2א0是实数的“连续性”的基数，CH说，每一组无数的实数都具有2א0。那个数学家对红衣主教的秩序有很好的处理，因此，在事实证明，试图证明CH的努力。 ZF]设定理论的公理。
One of the basic results in set theory is that the cardinality of the power set of the natural numbers is the same as the cardinality of the real numbers, which is strictly greater than the cardinality of the naturals. In fact Cantor proved a more general theorem: for any set X, the cardinality of X is strictly less than the cardinality of the power set of X. Since there is an infinite set, say the naturals, each application of the power set gives a greater infinite number. For each infinite set X we assign a cardinal number אX . It is equivalent to the Axiom of Choice that every set of alephs is linearly ordered. In fact, every set of cardinals is well ordered, so we can index the alephs with ordinals, α, β, in such a way that α < β iff אα < אβ. The problem is that we don’t know which cardinals go where in the ordering. We know that countably infinite sets have the smallest infinite cardinal, א0, and we know א0 is less than 2א0 , but we don’t know if there are any cardinals between. Cantor could not find any sets whose cardinalities were greater than א0 but less than 2א0 , so Cantor hypothesized that 2א0 is actually the next cardinal after א0, i.e. 2א0 = א1. This statement is known as the Continuum Hypothesis [CH] since one can prove that 2א0 is the cardinality of the “continuum” of real numbers, <. Equivalently, CH says that every uncountable set of real numbers has cardinality 2א0 . Proving CH would mean that mathematicians have a very good handle on the ordering of the cardinals, so much effort was spent trying to prove CH from the axioms of set theory. As it turns out, however, CH cannot be proven true or false from the Zermelo-Fraenkel [ZF] axioms of set theory. In other words, ZF cannot imply CH, and ZF cannot imply the negation of CH (if ZF is consistent to start with). Thus we say CH is independent of ZF in