从 5 出发的 Goodstein 序列，第三项是 255，在底数 4 下写成 $255=3\cdot 4^3+3\cdot 4^2+3\cdot 4+3$。把 4 换成 5 得 $3\cdot 125+3\cdot 25+3\cdot 5+3=468$，减 1，第四项是 $467$。

两种说法等价：${\aleph }_1$ 是比 ${\aleph }_0$ 大的第一个基数，所以“中间没有别的大小”就是“$2^{{\aleph }_0}$ 恰好是 ${\aleph }_1$”（这里用到选择公理，保证 $2^{{\aleph }_0}$ 是某个 $\aleph$）。

Cantor 1878 年就这样猜了，此后几十年一直想证明它，没有成功。1900 年，Hilbert 在巴黎的国际数学家大会上列了一串问题，第一个就是它。同一个问题里，Hilbert 还提到 Cantor 的另一个断言：实数能排成良序（第三讲），并请人给出直接的证明。后一半四年后由 Zermelo 用选择公理解决；前一半，就是这一讲的故事。

直觉上 CH 很可信：我们能具体写出来的实数集合，好像总是要么可数，要么和实数一样多。下面这条定理说，至少对闭集，这个直觉是对的。

这条定理的历史有点曲折：Cantor 1883 年先不加证明地写下一个版本，同年 Bendixson 在给他的信里指出其中一处不对，改正后给了证明；“完美集恰好有 $\mathfrak{c}$ 个点”是 Cantor 1884 年证的。1916 年，Alexandrov 和 Hausdorff 各自把结论推广到了更复杂的集合（Borel 集），它们也满足 CH。问题在于：实数的子集远不止这些“看得见”的集合。

关于 $2^{{\aleph }_0}$ 到底是多大，通常的公理系统 ZFC 能证明的东西少得可怜。最重要的一条来自 Julius König，背后还有一段插曲：1904 年他在国际数学家大会上宣布，实数根本不能排成良序，CH 因此没有意义。论证里的一个引理很快被发现有漏洞，第二年他在论文里明确收回了这个结论，只留下下面这条站得住的部分。

推论马上就来：$2^{{\aleph }_0}\ne {\aleph }_{\omega }$。因为 ${\aleph }_{\omega }$ 恰好是 ${\aleph }_0,{\aleph }_1,{\aleph }_2,\dots$ 这可数个更小基数的并。同样的理由排除 ${\aleph }_{\omega \cdot 2}$、${\aleph }_{{\omega }^2}$ 等所有“能由可数步逼近”的 $\aleph$。König 本人只证了这种可数步的情形；它的一般形式（无穷多个基数之和小于对应的积）是 Zermelo 1908 年写下的。

除此之外，ZFC 对 $2^{{\aleph }_0}$ 几乎什么都说不出来。后面会看到，这不是我们不够聪明。

“证明一个命题证不出来”听起来像绕口令。办法其实很朴素：找一个满足所有公理、却让这个命题为假的“世界”（模型）。如果命题能从公理推出来，它在所有满足公理的世界里都得成立。

1938 年 Gödel 造出了一个让 CH 成立的世界，叫“可构造宇宙” $L$。

$L$ 一层一层搭起来：$L_0$ 是空集；每一层 $L_{\alpha +1}$ 只收 $L_{\alpha }$ 里“能用一句一阶公式定义出来”的子集；到了极限序数就把前面的并起来。和全体集合比，$L$ 非常“吝啬”：它只放进能被说出来的集合。

另一半更难：造一个 CH 不成立的世界。$L$ 是把宇宙“削瘦”，Cohen 要做的是“加胖”，往宇宙里塞进新的实数。可塞什么？宇宙里已经有了“所有”实数，新实数从哪来？

Cohen 1963 年的办法叫力迫法（forcing）。先打个比方：在有理数里加进 $\sqrt{2}$，你不需要知道 $\sqrt{2}$“到底是什么”，只需要知道它满足什么方程，就能把扩张后的世界算清楚。力迫法是一样的思路：只用“有限的信息”去描述一个新实数，再证明这些信息能拼成一个和原来所有实数都不同的实数。

说得具体一点。从一个满足 ZFC 的可数模型 $M$ 出发。一个“条件” $p$ 是一个有限的 0/1 串，表示新实数的前几位；更长的串是更强的条件。$M$ 里的一个集合 $D$ 叫“稠密”，如果任何条件都能延长到 $D$ 里。

把 $g$ 加进 $M$，再把能由它造出来的集合都放进去，就得到更大的模型 $M[g]$。力迫法的核心定理说：$M[g]$ 仍然满足 ZFC，而且 $M[g]$ 里什么成立，可以在 $M$ 里面用“条件”算出来。

Cohen 把证明寄给 Gödel 检查；Gödel 读完写信称赞，并亲自把它推荐给《美国国家科学院院刊》发表。Gödel 加 Cohen：CH 和 ZFC 独立，既证不出来，也否定不了。Cohen 因此获得 1966 年的菲尔兹奖。2019 年，Han 和 van Doorn 在证明助手 Lean 里把这整套论证形式化了一遍。

Cohen 的方法不只能让 $2^{{\aleph }_0}={\aleph }_2$。换一换加入的新实数的个数，就能让它等于各种各样的 $\aleph$。到底哪些可以？

换句话说：除了“它比 ${\aleph }_0$ 大”，ZFC 对 $2^{{\aleph }_0}$ 的全部了解，就是定理 2 那一条。

“独立”回答了“能不能证”，没有回答“对不对”。如果你相信集合有一个确定的宇宙，那 CH 在里面要么真、要么假，只是 ZFC 的公理不够多，看不出来。Gödel 就这么想。1947 年他写道，CH 最可能是独立的，而他期待新的公理会否定它；晚年的一份手稿里，他甚至猜实数的个数是 ${\aleph }_2$。

有一个漂亮的等价形式，能让人对 CH 产生“几何直觉”。

1986 年 Freiling 把这个等价形式变成了一个思想实验。往实数轴上随机掷两支飞镖，得到 $x$、$y$。如果 CH 成立，按上面的分解，要么 $y$ 落在“$x$ 那条竖线上可数个坏点”里，要么 $x$ 落在“$y$ 那条横线上可数个坏点”里。可一个随机点落进一个事先给定的可数集，概率是 0。按“掷镖的直觉”，这两件事都不该发生，所以 CH 应该是假的。

这个论证的漏洞在于：那些“可数集”是由不可测的分解给出的，概率论在这里并不适用。所以它是直觉，不是证明，信不信由你。

今天的集合论学者在这个问题上分成几派。有人在找新的、大家都能接受的公理来决定 CH：一类强力的“力迫公理”推出 $2^{{\aleph }_0}={\aleph }_2$，2021 年 Asperó 和 Schindler 把其中两条主要的候选公理连了起来。另一条路线希望找到一个“终极的 $L$”，它会让 CH 成立。还有人认为根本没有唯一的宇宙，CH 在有的宇宙里真、有的宇宙里假，就像平行公设在不同的几何里一样。这是下一讲的话题。

这一讲的主角是一个答不出的问题。Cantor 猜 $2^{{\aleph }_0}={\aleph }_1$；闭集都满足它，这让猜想显得很可信；König 只能排除 ${\aleph }_{\omega }$ 这类位置；Gödel 造了一个瘦的宇宙让它成立，Cohen 造了一个胖的宇宙让它不成立。证明“证不出来”的那把钥匙，竟然又是第一讲那条对角线。

如果公理不够用，要不要加新公理？加什么？“更大的无穷”能不能帮上忙？下一讲，这个系列的最后一讲，往无穷的最高处走。

公理说不出的，
不一定没有答案；
也可能是问题本身有好几个答案。

In the Goodstein sequence starting from 5, the third term is 255, written in base 4 as $255=3\cdot 4^3+3\cdot 4^2+3\cdot 4+3$. Replace 4 by 5 to get $3\cdot 125+3\cdot 25+3\cdot 5+3=468$, subtract 1, and the fourth term is $467$.

The two forms say the same thing: ${\aleph }_1$ is the first cardinal larger than ${\aleph }_0$, so “no size in between” means exactly “$2^{{\aleph }_0}$ is ${\aleph }_1$” (this uses the axiom of choice, which guarantees that $2^{{\aleph }_0}$ is some $\aleph$).

Cantor made this conjecture in 1878 and spent decades afterwards trying to prove it, without success. In 1900, at the International Congress of Mathematicians in Paris, Hilbert presented a list of problems, and this was the first. In the same problem Hilbert also mentioned another of Cantor's claims, that the real numbers can be well-ordered (Lecture 3), and asked for a direct proof. Zermelo settled that second half four years later with the axiom of choice; the first half is the story of this lecture.

Intuitively, CH is very believable: every set of real numbers we can actually write down seems to be either countable or as large as the reals. The next theorem says that, for closed sets at least, this intuition is right.

The history of this theorem is a little tangled. In 1883 Cantor first stated a version without proof; the same year Bendixson pointed out, in a letter to him, that part of it was wrong, corrected it and gave a proof. That a perfect set has exactly $\mathfrak{c}$ points was proved by Cantor in 1884. In 1916 Alexandrov and Hausdorff independently extended the result to more complicated sets, the Borel sets, which satisfy CH as well. The trouble is that the subsets of the real line go far beyond these “visible” sets.

The usual axiom system, ZFC, can prove pitifully little about how big $2^{{\aleph }_0}$ really is. The most important result is due to Julius König, and there is a story behind it. At the 1904 International Congress of Mathematicians he announced that the real numbers cannot be well-ordered at all, so that CH would be meaningless. A lemma in his argument was soon found to have a gap, and the next year, in a paper, he explicitly withdrew that conclusion, keeping only the part that holds up, stated below.

A corollary follows at once: $2^{{\aleph }_0}\ne {\aleph }_{\omega }$. For ${\aleph }_{\omega }$ is exactly the union of the countably many smaller cardinals ${\aleph }_0,{\aleph }_1,{\aleph }_2,\dots$. The same reasoning rules out ${\aleph }_{\omega \cdot 2}$, ${\aleph }_{{\omega }^2}$ and every $\aleph$ that “can be approached in countably many steps.” König himself proved only this countable case; the general form (an infinite sum of cardinals is smaller than the corresponding product) was written down by Zermelo in 1908.

Beyond this, ZFC can say almost nothing about $2^{{\aleph }_0}$. As we will see, this is not because we are not clever enough.

“Proving that a statement cannot be proved” sounds like a tongue-twister. The method is actually quite plain: find a “world” (a model) that satisfies all the axioms but in which the statement is false. If the statement could be derived from the axioms, it would have to hold in every world that satisfies them.

In 1938 Gödel built a world in which CH holds, called the “constructible universe” $L$.

$L$ is built level by level: $L_0$ is the empty set; each level $L_{\alpha +1}$ takes only those subsets of $L_{\alpha }$ that “can be defined by a first-order formula”; at limit ordinals, take the union of all the levels before. Compared with the universe of all sets, $L$ is very “stingy”: it lets in only the sets that can be put into words.

The other half is harder: building a world in which CH fails. $L$ makes the universe “thinner”; Cohen had to make it “fatter,” stuffing new real numbers into the universe. But stuff in what? The universe already contains “all” the real numbers, so where would a new one come from?

Cohen's method of 1963 is called forcing. An analogy first: to add $\sqrt{2}$ to the rational numbers, you don't need to know what $\sqrt{2}$ “really is”; it is enough to know which equation it satisfies, and you can work out the extended world completely. Forcing follows the same idea: describe a new real using only “finite pieces of information,” then prove that these pieces fit together into a real different from every real already there.

To be more concrete: start from a countable model $M$ of ZFC. A “condition” $p$ is a finite string of 0s and 1s, standing for the first few digits of the new real; a longer string is a stronger condition. A set $D$ in $M$ is called “dense” if every condition can be extended to one in $D$.

Add $g$ to $M$, together with every set that can be built from it, and you get a larger model $M[g]$. The central theorem of forcing says that $M[g]$ still satisfies ZFC, and that what holds in $M[g]$ can be worked out inside $M$ using the “conditions.”

Cohen sent his proof to Gödel to check; Gödel read it, wrote back in praise, and personally recommended it to the Proceedings of the National Academy of Sciences for publication. Gödel plus Cohen: CH is independent of ZFC, and can be neither proved nor refuted. For this Cohen received the Fields Medal in 1966. In 2019 Han and van Doorn formalized the whole argument in the proof assistant Lean.

Cohen's method can do more than make $2^{{\aleph }_0}={\aleph }_2$. By varying the number of new reals added, you can make it equal to all sorts of $\aleph$. Which ones, exactly?

In other words: apart from “it is larger than ${\aleph }_0$,” everything ZFC knows about $2^{{\aleph }_0}$ is that one fact in Theorem 2.

“Independent” answers the question “can it be proved?” and not the question “is it true?” If you believe that there is one definite universe of sets, then CH is either true or false in it; the axioms of ZFC are simply too few to tell which. That is what Gödel thought. In 1947 he wrote that CH was most likely independent, and that he expected new axioms to refute it; in a manuscript from late in his life he even conjectured that the number of real numbers is ${\aleph }_2$.

There is a lovely equivalent form that gives CH a kind of “geometric intuition.”

In 1986 Freiling turned this equivalent form into a thought experiment. Throw two darts at random at the real line, getting $x$ and $y$. If CH holds, then by the decomposition above, either $y$ lands among “the countably many bad points on the vertical line through $x$,” or $x$ lands among “the countably many bad points on the horizontal line through $y$.” But a random point lands in a countable set fixed in advance with probability 0. By “dart-throwing intuition,” neither should happen, so CH should be false.

The flaw in this argument is that those “countable sets” come from a non-measurable decomposition, where probability theory does not apply. So it is an intuition, not a proof; believe it or not, as you like.

Set theorists today are divided on this question. Some are looking for new axioms that everyone can accept and that decide CH: a family of strong “forcing axioms” implies $2^{{\aleph }_0}={\aleph }_2$, and in 2021 Asperó and Schindler connected two of the main candidate axioms. Another line of work hopes to find an “ultimate $L$,” which would make CH true. Still others think there is no single universe at all: CH is true in some universes and false in others, much like the parallel postulate across different geometries. That is the topic of the next lecture.

The main character of this lecture is a question with no answer. Cantor conjectured $2^{{\aleph }_0}={\aleph }_1$; all closed sets satisfy it, which makes the conjecture look very believable; König could rule out only positions like ${\aleph }_{\omega }$; Gödel built a thin universe in which it holds, and Cohen a fat universe in which it fails. And the key to proving “it cannot be proved” turned out to be, once again, the diagonal from Lecture 1.

If the axioms are not enough, should we add new ones? Which ones? Can “bigger infinities” help? The next lecture, the last in this series, climbs to the very top of infinity.

What the axioms cannot decide
need not lack an answer;
the question itself may have several.