只看它能不能由可数步逼近。${\aleph }_{\omega \cdot 2}$ 是 ${\aleph }_{\omega +1},{\aleph }_{\omega +2},\dots$ 的极限，不行；${\aleph }_{{\omega }_1+\omega }$ 是 ${\aleph }_{{\omega }_1+1},{\aleph }_{{\omega }_1+2},\dots$ 的极限，也不行。${\aleph }_{{\omega }_2+1}$ 是后继基数，可以；${\aleph }_{{\omega }_1}$ 要逼近它得走 ${\omega }_1$ 步，可数步不够，也可以。

先回到最底下。ZFC 的公理能造出多大的集合？从空集出发，反复取幂集，再把前面的都并起来，得到一层一层的“累积层级”：

最后一条用于极限序数 $\lambda$。

$V_1,V_2,V_3,V_4,V_5$ 分别有 1、2、4、16、65536 个元素。$V_{\omega }$ 是所有“有限的”集合；$V_{\omega +1}$ 已经包含了全体实数。所有集合都在某一层里。

什么样的 $\kappa$ 算“够大”？要求是：从比它小的东西出发，用 ZFC 允许的任何操作，都造不出它来。

${\aleph }_0$ 满足这两条（有限个有限数的并还是有限，有限集的幂集还是有限），所以不可达基数就是“不可数版的 ${\aleph }_0$”：从下面怎么够都够不着。1908 年 Hausdorff 第一个问这种数存不存在，说这个问题“只能悬而未决”；1930 年 Sierpiński 和 Tarski 给了上面的定义，“不可达”这个名字是 Kuratowski 起的。

这条定理有一个惊人的推论：

这就是梯子的规则：每一级大基数的存在，都推出下面一级的系统不矛盾，因而不可能在下面一级里被证明。爬梯子不是“证明”出来的，只能“假设”上去。

梯子上更高的一级，来自一个很具体的问题：能不能给实数的所有子集定义“长度”？要求可数可加、单点长度为 0、$[0,1]$ 长度为 1。Lebesgue 测度做不到所有子集（1905 年 Vitali 就造出过不可测集，用的正是平移不变性）；但如果不要求平移不变呢？

1930 年 Ulam 研究了这个问题，发现答案取决于基数有多大。最干净的一步是这样的：

推论：如果连续统假设成立，$[0,1]$ 和 ${\omega }_1$ 一样多，测度可以搬过去，所以 $[0,1]$ 的所有子集上也不存在这样的测度（这一点 Banach 和 Kuratowski 1929 年已经证明）。Ulam 把连续统假设这个前提去掉了：要想有这样的测度，$2^{{\aleph }_0}$ 至少要大到一个（弱）不可达基数。

把测度换成只取 0 和 1 两个值的，就得到“可测基数”：基数 $\kappa$ 的所有子集上有一个只取 0 和 1、单点为 0、对少于 $\kappa$ 个集合的不交并都可加的测度。Ulam 和 Tarski 证明这样的 $\kappa$ 必定是不可达的；1960 年代 Hanf、Keisler 和 Tarski 又证明，它远在最小的不可达基数之上。

换句话说，可测基数一旦存在，上一讲那个“瘦宇宙”就不可能是全部。从这一级往上，大基数开始和宇宙的“形状”有关了：一个可测基数等价于存在一个从 $V$ 到某个子宇宙 $M$ 的“嵌入” $j$，它保持所有一阶性质，只把 $\kappa$ 往上挪。梯子再往上的几级，都是要求这个嵌入更“贴近”恒等。

这就是梯子的顶。1967 年 Reinhardt 在博士论文里提出“存在 $j:V\to V$”这种最强的公理，Kunen 证明它和选择公理矛盾。梯子在这里撞上了天花板，或者说，撞上了选择公理：不承认选择公理时，它至今还没被驳倒。

一个连自己存在都证不出的公理，凭什么值得相信？最有力的理由是：它们对“下面”的世界有可检验的后果，而且这些后果彼此一致、相当漂亮。最好的例子是无穷博弈。

对“简单”的 $A$，比如开集，这个博弈总是“决定的”：总有一方有必胜策略。但不是所有的 $A$ 都这样：

这和第四讲 Sierpiński 的分解一样，是借一个良序“一个个对付过去”的论证。它造出的集合非常“病态”。那么，对所有能具体描述出来的集合，博弈都是决定的吗？

这是大基数最漂亮的成绩之一：ZFC 本身证明不了“所有射影集都可测”，加上 Woodin 基数以后，一整片关于实数的问题都有了答案，而且答案都是“好”的那一边。反过来，射影集的决定性也推出 Woodin 基数的某种相容性，两者互相印证。

那么，大基数能不能也回答连续统假设？

Gödel 1947 年其实已经料到了。他说集合的概念本身就提示着新的“无穷公理”，比如不可达基数，加进去并不武断；但他也指出，他构造 $L$ 的论证对这些公理照样成立，所以靠它们解决连续统问题希望不大，得另找基于全新原理的公理。大基数确实解决了很多问题，偏偏绕开了这一个。

CH 独立以后，集合论学者对“它到底对不对”分成了两大派。

一个宇宙。集合有一个确定的宇宙，我们的任务是找到描述它的正确公理。大基数之所以可信，是因为它们的后果彼此一致，就像一个物理理论被不同的实验从不同方向验证。Woodin 多年来在找这样的公理：早年他论证 CH 应该是假的；后来他转向一个叫“终极 $L$”的构想，它会让 CH 成立。

多宇宙。Hamkins 2012 年系统地提出另一种看法：根本没有唯一的宇宙。（早在 1930 年，Zermelo 就设想过一串没有尽头的“集合论宇宙”，每一个都是下一个里面的一个集合。）每个满足 ZFC 的模型都是一个真实的“世界”，力迫法让我们在世界之间穿梭。CH 在有的世界里真、有的世界里假，问“它到底对不对”，就像问平行公设到底对不对：在欧氏几何里对，在双曲几何里不对，两种几何都是好的数学。

这两派还在争。它们共同的出发点，是 Cantor 1883 年为他的超限数辩护时写下的一句话：

只要不矛盾，数学家就有权定义新的对象、研究它们。超限数当年被当成异端；今天，连“比 ZFC 能证明存在的一切都大”的无穷，也只是一架梯子上的又一级。

这个系列从“正整数和偶数谁多”出发，一路走到了一个至今没有答案的问题。贯穿始终的工具只有一个：对角线。第一讲用它证明实数数不完；第四讲 Cohen 用它造出新的实数；这一讲 Gale–Stewart 的论证里，它又换成良序的样子出现了一次。一个一百多年前的小技巧，到今天还在撑着整门学科。

无穷的大小，到今天也没有被数完。也许，这就是它值得一直数下去的原因。

梯子没有顶；
爬上去的每一级，
都让下面看得更清楚。

All that matters is whether it can be approached in countably many steps. ${\aleph }_{\omega \cdot 2}$ is the limit of ${\aleph }_{\omega +1},{\aleph }_{\omega +2},\dots$, so no; ${\aleph }_{{\omega }_1+\omega }$ is the limit of ${\aleph }_{{\omega }_1+1},{\aleph }_{{\omega }_1+2},\dots$, so no again. ${\aleph }_{{\omega }_2+1}$ is a successor cardinal, so yes; approaching ${\aleph }_{{\omega }_1}$ takes ${\omega }_1$ steps, and countably many are not enough, so yes too.

Back to the bottom first. How large are the sets that the axioms of ZFC can build? Start from the empty set, take power sets again and again, and take the union of everything so far; this gives the level-by-level “cumulative hierarchy”:

The last line is for limit ordinals $\lambda$.

$V_1,V_2,V_3,V_4,V_5$ have 1, 2, 4, 16 and 65,536 elements respectively. $V_{\omega }$ consists of all the “finite” sets; $V_{\omega +1}$ already contains all the real numbers. Every set lies in some level.

What kind of $\kappa$ counts as “big enough”? The requirement is this: starting from things smaller than it, no operation that ZFC allows can build it.

${\aleph }_0$ satisfies both conditions (a union of finitely many finite numbers is finite, and the power set of a finite set is finite), so an inaccessible cardinal is an “uncountable version of ${\aleph }_0$”: however you reach for it from below, you cannot get there. In 1908 Hausdorff was the first to ask whether such numbers exist, and said that the question “can only be left open”; in 1930 Sierpiński and Tarski gave the definition above, and the name “inaccessible” is due to Kuratowski.

This theorem has a startling consequence:

This is the rule of the ladder: the existence of each rung of large cardinals implies that the system one rung below is consistent, and so it cannot be proved there. You don't get up the ladder by “proving”; you can only get up by “assuming.”

A higher rung of the ladder comes from a very concrete question: can we define a “length” for all subsets of the real line? We want it to be countably additive, to give each single point length 0, and to give $[0,1]$ length 1. Lebesgue measure cannot do this for all subsets (Vitali built a non-measurable set back in 1905, using precisely translation invariance); but what if we don't require translation invariance?

Ulam studied this question in 1930 and found that the answer depends on how large the cardinal is. The cleanest step goes like this:

A corollary: if the continuum hypothesis holds, then $[0,1]$ and ${\omega }_1$ are the same size, the measure can be carried over, and so there is no such measure on all subsets of $[0,1]$ either (Banach and Kuratowski had already proved this in 1929). Ulam removed the assumption of the continuum hypothesis: for such a measure to exist, $2^{{\aleph }_0}$ must be at least as large as a (weakly) inaccessible cardinal.

Replace the measure by one that takes only the values 0 and 1, and you get “measurable cardinals”: a cardinal $\kappa$ is measurable if there is a measure on all its subsets that takes only the values 0 and 1, is 0 on single points, and is additive over disjoint unions of fewer than $\kappa$ sets. Ulam and Tarski proved that such a $\kappa$ must be inaccessible; in the 1960s Hanf, Keisler and Tarski went on to show that it lies far above the least inaccessible cardinal.

In other words, once a measurable cardinal exists, the “thin universe” of the last lecture cannot be everything. From this rung up, large cardinals begin to bear on the “shape” of the universe: a measurable cardinal is equivalent to an “embedding” $j$ from $V$ into some sub-universe $M$ that preserves every first-order property and only moves $\kappa$ upward. The rungs further up all ask for this embedding to be ever “closer” to the identity.

This is the top of the ladder. In his 1967 doctoral thesis Reinhardt proposed the strongest axiom of this kind, “there exists $j:V\to V$,” and Kunen proved that it contradicts the axiom of choice. Here the ladder hits the ceiling, or rather, it hits the axiom of choice: without the axiom of choice, it has not been refuted to this day.

Why should we believe an axiom when what it asserts cannot even be proved to exist? The strongest reason is that these axioms have testable consequences for the world “below,” and those consequences agree with one another and are rather beautiful. The best example is infinite games.

For “simple” $A$, such as open sets, this game is always “determined”: one side or the other has a winning strategy. But not every $A$ is like this:

Like Sierpiński's decomposition in Lecture 4, this is an argument that uses a well-ordering to “deal with them one at a time.” The set it produces is extremely “pathological.” So is the game determined for every set that can be concretely described?

This is one of the finest achievements of large cardinals. ZFC by itself cannot prove that “every projective set is measurable,” but once Woodin cardinals are added, a whole swathe of questions about the real numbers gets answered, and the answers all come down on the “nice” side. Conversely, projective determinacy also implies a certain consistency of Woodin cardinals, so each confirms the other.

So can large cardinals answer the continuum hypothesis too?

Gödel had in fact foreseen this in 1947. He said that the very concept of set suggests new “axioms of infinity,” such as inaccessible cardinals, and that adding them is not arbitrary; but he also pointed out that his argument constructing $L$ works just as well with these axioms, so there was little hope of settling the continuum problem with them, and axioms based on entirely new principles would be needed. Large cardinals have indeed settled many problems, but this one, of all problems, slips past them.

After CH was shown to be independent, set theorists split into two main camps over “is it true or not?”

One universe. There is one definite universe of sets, and our task is to find the right axioms describing it. Large cardinals are credible because their consequences agree with one another, much as a physical theory is confirmed by different experiments from different directions. Woodin has spent many years looking for such axioms: early on he argued that CH should be false; later he turned to a proposal called “Ultimate $L$,” which would make CH true.

A multiverse. In 2012 Hamkins set out another view systematically: there is no unique universe at all. (As early as 1930, Zermelo had imagined an endless sequence of “set-theoretic universes,” each one a set inside the next.) Every model of ZFC is a real “world,” and forcing lets us travel between worlds. CH is true in some worlds and false in others, and asking “is it really true?” is like asking whether the parallel postulate is really true: it is true in Euclidean geometry and false in hyperbolic geometry, and both geometries are good mathematics.

The two camps are still arguing. Their common starting point is a sentence Cantor wrote in 1883, defending his transfinite numbers:

As long as there is no contradiction, mathematicians have the right to define new objects and study them. The transfinite numbers were once treated as heresy; today even an infinity “bigger than everything ZFC can prove to exist” is just one more rung on a ladder.

This series set out from “are there more positive integers or more even numbers?” and has ended at a question that still has no answer. One tool ran through it from start to finish: the diagonal. Lecture 1 used it to show that the real numbers cannot be counted; in Lecture 4 Cohen used it to build new real numbers; and in this lecture it turned up once more, in the guise of a well-ordering, in the Gale–Stewart argument. A little trick from more than a century ago is still holding up an entire field.

To this day, no one has finished counting the sizes of infinity. Perhaps that is exactly why it is worth going on counting.

The ladder has no top;
each rung you climb
shows you more clearly what lies below.