The continuum hypothesis in intuitionism
The continuum hypothesis in intuitionism
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直觉主义的连续统假说
DOI：
10.2307/2273264
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发表时间：
1981
期刊：
影响因子：
--
通讯作者：
W. Veldman
中科院分区：
文献类型：
--
作者：
W. Gielen;H. Swart;W. Veldman
尽管布劳威尔因其对经典逻辑和集合论的激烈攻击而闻名，但他的工作并不是在真空中发展的，而是强烈依赖于康托尔的工作。他一心想抛开非建设性的论点，试图沿着新的、直觉主义的路线重建康托的大厦。集合论核心的连续统假说也面临着布劳威尔的挑战，他不得不面对康托在试图解决它时所能得出的最远的结论：实线的每一个不可数闭子集都具有连续统的力量。布劳威尔对此的思考似乎有了一些发展。 1914 年，我们听到他说：“Wir sahen oben dass das Cantorsche Haupttheorem für den Intuitionisten keines Beweises bedarf”（“正如我们在上面看到的，对于我们来说，作为直觉主义者，康托主定理不需要证明”）[3]。尽管如此，五年后，他发表了一篇文章：Theorie der Punktmengen，这可以被描述为详细重建康托推理的尝试[4]。正如布劳威尔在 1952 年承认的那样，这一尝试并不完全成功，他现在可能已经失去了一些年轻时的鲁莽 [10]。那么康托尔大定理的建设性内容是什么，这个问题仍然有待解答。我们认为我们给出的答案不能被视为结论性的答案，但无论如何，这都是一个开始。
Although Brouwer became famous for his vehement attacks upon classical logic and set theory, his work did not develop in a vacuum and strongly depended on that of Cantor. His mind bent on shifting aside nonconstructive arguments, he tried to rebuild Cantor's edifice along new, intuitionistic lines. The continuum hypothesis, lying at the core of set theory, also confronted Brouwer, and he had to face the farthest conclusion Cantor had been able to reach in trying to solve it: every nondenumerable closed subset of the real line has the power of the continuum. Brouwer's thinking about it seems to have been subject to some development. In 1914 we hear him saying: “Wir sahen oben dass das Cantorsche Haupttheorem für den Intuitionisten keines Beweises bedarf” (“As we saw above, for us, being intuitionists, Cantor's Main Theorem does not need a proof”) [3]. Nevertheless, five years later, he publishes an essay: Theorie der Punktmengen, which might be described as an attempt to reconstruct Cantor's reasonings in detail [4]. This attempt was not entirely successful, as Brouwer comes to admit in 1952, probably having lost, now, some of his youthful rashness [10]. So the question of what the constructive content of Cantor's Main Theorem is, still awaits an answer. We do not think the answer we will give can be considered a conclusive one, but, in any case, it is a beginning.