陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「A computational perspective on set theory」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

The standard modern foundation of mathematics is constructed using set theory . With these foundations, the mathematical universe of objects one studies contains not only the “primitive” mathematical objects such as numbers and points, but also sets of these objects, sets of sets of objects, and so forth. (In a pure set theory, the primitive objects would themselves be sets as well; this is useful for studying the foundations of mathematics, but for most mathematical purposes

One may suspect that the reason for this unintuitive behaviour is the presence of infinite sets in one’s mathematical universe. After all, if one deals solely with finite sets, then there is no need to distinguish between countable and uncountable infinities, and Banach-Tarski type paradoxes cannot occur.

已知结果和反例

On the other hand, many statements in infinitary mathematics can be reformulated into equivalent statements in finitary mathematics (involving only finitely many points or numbers, etc.); I have explored this theme in a number of previous blog posts . So, one may ask: what is the finitary analogue of statements such as Cantor’s theorem or the Banach-Tarski paradox?

The finitary analogue of Cantor’s theorem is well-known: it is the assertion that for every natural number , or equivalently that the power set of a finite set of elements cannot be enumerated by itself. Though this is not quite the end of the story; after all, one also has for every natural number , or equivalently that the union of a finite set and an additional element cannot be enumerated by itself, but the former statement extends to the infinite case, while the latter o

证明或构造的主线

On the other hand, it is less obvious what the finitary version of the Banach-Tarski paradox is. Note that this paradox is available only in three and higher dimensions, but not in one or two dimensions; so presumably a finitary analogue of this paradox should also make the same distinction between low and high dimensions.

I therefore set myself the exercise of trying to phrase Cantor’s theorem and the Banach-Tarski paradox in a more “finitary” language. It seems that the easiest way to accomplish this is to avoid the use of set theory, and replace sets by some other concept. Taking inspiration from theoretical computer science, I decided to replace concepts such as functions and sets by the concepts of algorithms and oracles instead, with various constructions in set theory being replaced inst

阅读时建议盯住的点

I describe this viewpoint, and how statements such as Cantor’s theorem and Banach-Tarski are interpreted with this viewpoint, below the fold. I should caution that this is a conceptual exercise rather than a rigorous one; I have not attempted to formalise these notions to the same extent that set theory is formalised. Thus, for instance, I have no explicit system of axioms that algorithms and oracles are supposed to obey. Of course, these formal issues have been explored in g

A second caveat is that the actual semantic content of this post is going to be extremely low. I am not going to provide any genuinely new proof of Cantor’s theorem, or give a new construction of Banach-Tarski type; instead, I will be reformulating the standard proofs and constructions in a different language. Nevertheless I believe this viewpoint is somewhat clarifying as to the nature of these paradoxes, and as to how they are not as fundamentally tied to the nature of sets

值得单独记下的条目

• takes a real number as input.
• If has previously answered the question of whether lies in , repeat whatever answer was given previously.
• If has not been queried before, and if furthermore no rational translate of has been queried before either (i.e. differs from all previous queries by an irrational number), then answer with “yes”.
• Finally, if has not been queried before, but has been queried before for some rational , then answer with “no”.
• Store (and ) in memory for use in future queries.
• A description of an elementary set (which can be done in a finite amount of time and space, by specifying the number of intervals used to form , together with their endpoints);
• An interval oracle , that for each natural number input returns an interval ;
• A covering oracle , which for each real number in , returns a natural number for which . (Actually, this oracle is redundant, as it can be simulated from the previous two outputs by a finite brute force search.)

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源