The n-Category Café

Re: Axiomatic Set Theory 8: Well Ordered Sets

I’ve read someone jokingly say Cantor was studying convergence of Fourier series and did such a bad job he wound up inventing infinite ordinals. His focus on pointwise convergence of Fourier series looks a bit wrongheaded today: it’s good if you like really hard problems, but everything gets easier if you use $L^1$ or $L^2$ convergence.

From Nicholas Scoville’s Georg Cantor at the dawn of point-set topology:

Heine’s 1870 work showed that if a function is almost everywhere continuous and its trigonometric series converges uniformly, then the Fourier series is unique. As Dauben points out,

Requiring almost-everywhere continuity and uniform convergence, Heine’s theorem invited direct generalizations.

These generalizations would be taken up by Cantor. Important for our purposes is that Cantor developed a proof technique in his 1870 paper and modified it only slightly while weakening his hypotheses in the 1872 paper. More specifically, Cantor showed that, under certain hypotheses, the trigonometric representation of a function remains unique even when convergence or representation of the function is given up on certain infinite subsets of the open interval (0,2π).

Cantor defined the derived set $P'$ to be the set of limit points of a set $P \subseteq \mathbb{R}$: in other words, $P$ minus its set of isolated points.

He then recursively defined the $\alpha$th derived set $P^{(\alpha)}$ by

$$P^{(0)} = P, \quad P^{(\alpha + 1)} = (P^{(\alpha)})^'$$

and said $P \subseteq \mathbb{R}$ is a set of the $\alpha$th kind if $P^{(\alpha + 1)}$ is empty.

It seems Cantor first showed this: if

$$\lim_{N \to \infty} \sum_{n = -N}^N c_n e^{2\pi n x} = 0$$

pointwise, then $c_n = 0$ for all $n$. This is a result about the uniqueness of Fourier series.

Then he got the same conclusion assuming the limit is zero except at a set of the first kind — i.e., except at a finite set of points.

Then he got the same conclusion assuming the limit is zero except at a set of the $n$th kind for any $n \in \mathbb{N}$.

But he didn’t want to stop there! So he defined ordinals, and for limit ordinals $\alpha$ he defined the $\alpha$th derived set by

$$P^{(\alpha)} = \bigcap_{\beta \lt \alpha} P^{(\beta)}$$

I want to know if Cantor succeeded in showing this: if

$$\lim_{N \to \infty} \sum_{n = -N}^N c_n e^{2\pi n x} = 0$$

except at a set of the $\alpha$th kind for some countable ordinal $\alpha$, does that imply $c_n = 0$ for all $n$? That seems to be where his work was heading, but I don’t even know if it’s true!

Now I’m curious about how far Cantor got in this direction — he may have gotten distracted by studying ordinals and cardinals for their own sake.

According to S. M. Srivasta, Cantor did show that for every $P \subseteq \mathbb{R}$ there exists a countable ordinal $\alpha$ such that for all $\beta \ge \alpha$ we have

$$P^{(\beta)} = P^{(\alpha)}$$

So there’s no point in studying the $\alpha$th derived set of $P \subseteq \mathbb{R}$ for uncountable ordinals $\alpha$.

Srivasta shows that for every countable closed set, $P^{(\alpha)} = \emptyset$ for some countable ordinal $\alpha$. Srivasta also says it “seems likely” that Cantor realized that if

$$\lim_{N \to \infty} \sum_{n = -N}^N c_n e^{2\pi n x} = 0$$

except for $x$ in a countable closed set, then $c_n = 0$ for all $n$.

That’s all I could quickly find about this, except: there exist sets of Lebesgue measure zero $P \subseteq \mathbb{R}$ such that

$$\lim_{N \to \infty} \sum_{n = -N}^N c_n e^{2\pi n x} = 0$$

except on $P$, yet we don’t have $c_n = 0$ for all $n$.

Re: Axiomatic Set Theory 8: Well Ordered Sets

I think there might be a typo near the start of the proof of Proposition 8.4.3 - $\preccurlyeq$ rather than $\leq$. It’s invoking Hartogs, which only supplies the latter. And later, you invoke the property that any function $W\to X$ can’t be injective, but how it was phrased was slightly confusing, because it seems like you were using the fact the function $g$ was order-preserving (esp. paired with the typo), when it’s not specific to $g$.