科学与数学quantamagazine.org
实数究竟有多少个？新证明让答案更近一步
How many real numbers exist? New proof moves closer to an answer
2021 — 2025，它一次次回到首页
黑点是上首页（≥ 50 分）的那次投稿，空心点是没冲上去的投稿，点越大分数越高。
导读
这篇 Quanta 文章围绕连续统问题展开：实数究竟有多少？在连续统假设被证明独立于 ZFC 的背景下，它介绍力迫法等工具，以及数学家如何通过寻找更自然的新公理（文中涉及 Woodin 的 Ultimate L 等）来逼近答案。
为什么经久不衰：它把“连续统假设是否独立”这一抽象难题，与数学家如何论证和选择新公理讲得可读，既硬核又触及数学哲学，所以隔几年仍值得重读。
HN 评论精选
补充力迫法入门和选择公理的哲学讨论，推荐进一步阅读。
好文章。看起来很多人感兴趣，我补充点背景。首先，力迫法是什么？文章对超幂（构造的关键部分）描述很好但稍快，推荐 Tim Chow 的《力迫法入门》，它不仅高层交代数学细节，还清楚解释证明某命题独立于公理的哲学。其次，文章提到数学家如何选择公理；Penelope Maddy 的《Believing the Axioms》是经典，聚焦集合论，描述马丁公理及正反论证。这些论证有时是审美（“规则”对“惊喜”），有时是实用（能证明什么），有时涉及宇宙应该怎样的深层价值。
pavpanchekha查看英文原文
Great article. Since it looks like a lot of folks are interested in this article, some extra background. First, what is forcing? The article actually has a great description of ultrapowers (a key part of the construction) but it goes by a little fast, so you might like Tim Chow's "A beginner's guide to forcing" [1] which does a good job not only laying out the mathematical details at a high level, but also really clearly explaining the philosophy of how you prove something independent of the axioms. I found it very illuminating. Second, the article has some interesting notes on how mathematicians go about what axioms to select and which not to. Penelope Maddy's "Believing the Axioms" [2] is the classic on this topic (it has two parts). It is focused on set theory, so it has a nice description of Martin's axiom and the arguments for and against. It was nice to read because it is a deeply technical argument (set theory is hard!) but at the same time there is no "right answer"—all of these axioms, after all, are independent of ZFC, and it's "ok" to add any of them. The arguments are sometimes aesthetic ("rules" versus "surprises"), sometimes pragmatic (what they can and cannot prove), and sometimes involve deep values of what the universe should be like (should higher cardinalities be like lower ones? weirder? simpler?). It might all seem abstract, but if your day job is programming, imagine an argument over how to architect a large and complex system. Perhaps both architectures are possible, but which one is "right"? What arguments would you deploy? Set theorists are also building a large and complex system (the universe of sets) and are having arguments over how it should be built, which things it should make easy and which hard, which technologies should be supported natively (forcing?) and which should not. [1] http://timothychow.net/forcing.pdf [2] https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf
提醒文章隐含柏拉图主义预设，非柏拉图主义者需换一种方式理解。
文章里有不少干货（结尾简短提到 Woodin 的 Ultimate L 计划）。我想指出，因为 Hacker News 读者大多不是数学柏拉图主义者，整篇文章隐含假设了柏拉图主义哲学基础：数学对象客观存在，每个逻辑公理都有真值，不能任意选择，选错会选到“错误”公理。因此连续统假设是否真仍重要，即使它独立于 ZFC，最终需要哲学而非数学论证。如果你不是柏拉图主义者，可把文中“真/假”理解为“与检验逻辑理论语义的元模型一致/不一致”。
dwohnitmok查看英文原文
Man there's a lot of juicy stuff in this article (Woodin's Ultimate L program gets briefly alluded to at the end of the article). I just want to point out, because the HN crowd seems to generally not be mathematical Platonists, that this entire article is implicitly assuming a Platonist philosophical foundation. This may cause confusion for lay readers who are not mathematical Platonists. In other words the article assumes that mathematical objects have an objective existence: they either exist or they do not. Hence every single logical axiom has a truth value. You do not get to arbitrarily choose what logical axioms you want. If you do, you can end up choosing the "wrong one." Therefore it is an important question to understand whether the Continuum Hypothesis is true or not, even if it's independent of ZFC (and hence requires ultimately philosophical rather than mathematical arguments). If you aren't a Platonist and instead view logical axioms as having no inherent truth value, but rather foundations that you can pick and choose from as necessary (where in one case you may choose to use an axiom and in another you may choose its negation), then all that might sound very strange to you. In that case, you should mentally substitute every instance of "true" or "false" in the article with "agrees or disagrees with the meta model used to examine the semantics of a logical theory." In particular, whenever we talk about a formal treatment of the semantics (i.e. model) of a theory, whether that be something like ZFC or a programming language, we must always make those statements relative to a meta-model. For example if we talk about the formal semantics of a language like C, we must first posit a meta-model which already contains notions of things like "integer" and "natural number" which can be used to give meaning to statements such as "performed an operation n times." If you're not a Platonist then you probably believe that there are multiple possible meta-models you could use.
连续统假设独立于 ZFC 是既定事实，新工作是在寻找自然新公理，标题措辞有误导。
50 年来数学家认为实数总数不可知。连续统假设独立于 ZFC 是既定结果，任何证明都不会改变这一点。所以这里证明的东西不改变独立性。Raymond Smullyan 说过，我们不是在找连续统假设的证明或否证，而是在找自然到足以当作公理的假设。文章小标题似乎概念不严谨；虽然正文后续严谨，但开头“没人知道”仍混淆了问题。
joe_the_user查看英文原文
For 50 years, mathematicians have believed that the total number of real numbers is unknowable. It's an established result that the Continuum Hypothesis is independent of Zermelo–Fraenkel axioms of set theory. No proof is going change that. So whatever has been proved here doesn't change that. It will take a second to get the reference but Raymond Smullyan says essentially that "we're not looking for a proof or disproof of CH, we're looking for an assumption that can be shown to be natural enough that we can take it as an axiom". Just sayin' since the (subheading) writer seems to be playing fast and loose with the concepts involved. Edit: article goes on to give rigorous explanation but still starting "no one knew" confuses what's going on.
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