- SET THEORY 545
His object was to show that F(x) must be a linear function, so that G(x) — F(x) —
Ax — Â — \a.QX^2 would be a quadratic polynomial that was also the sum of a
uniformly convergent trigonometric series, and hence itself a constant, from which
it would follow, first that an = 0, and then that all the other a„ and all the bn are
zero. To that end, he showed that its generalized second derivative
F(x + h) + F(x-h)-2F(x)
F
*(x)=
is& j?—-—
was zero wherever the original series converged to zero.^5 A weaker theorem, similar
to the theorem that a differentiable function must be continuous, implied that
Hm (F(x + h)-F(x)) + (F(x-h)-F(x)) = Q
h->0 hThe important implication of this last result is that the function F(x) cannot have a
comer. If it has a right-hand derivative at a point, it also has a left-hand derivative
at the point, and the two one-sided derivatives are equal. This fact, which at first
sight appears to have nothing to do with set theory, was a key step in Cantor's
work.
2.2. Cantor's work on trigonometric series. In 1872 Cantor published his
first paper on uniqueness of trigonometric series, finishing the proof that Riemann
had set out to give: that a trigonometric series that converges to zero at every point
must have all its coefficients equal to zero. In following the program of proving that
F(x) is linear and hence constant, he observed that it was not necessary to assume
that the series converged to zero at every point. A finite number of exceptions could
be allowed, at which the series either diverged or converged to a nonzero value. For
F(x) is certainly continuous, and if it is linear on [a, b] and also on [b, c], the fact
that it has no corners implies that it must be linear on [o, c]. Hence any isolated
exceptional point b could be discounted.
The question therefore naturally arose: Can one allow an infinite number of
exceptional points? Here one comes up against the Bolzano-Weierstrass theorem,
which asserts that the exceptional points cannot all be isolated. They must have
at least one point of accumulation. But exceptional points isolated from other
exceptional points could be discounted, just as before. That left only their points
of accumulation. If these were isolated—in particular, if there were only finitely
many of them—the no-corners principle would once again imply uniqueness of the
series.
Ordinal numbers. Cantor saw the obvious induction immediately. Denoting the set
of points of accumulation of a set Ñ (what we now call the derived set) by P', he
knew that P' D P" D P'" D · · ·. Thus, if at some finite level of accumulation points
of accumulation points a finite set was obtained, the uniqueness theorem would
remain valid. But the study of these sets of points of accumulation turned out to
be even more interesting than trigonometric series themselves. No longer dealing
with geometrically regular sets, Cantor was delving into point-set topology, as we
now call it. No properties of a geometric nature were posited for the exceptional
points he was considering, beyond the assumption that the sequence of derived sets
must terminate at some finite level. Although the points of any particular set (as
(^5) Hermann Amandus Schwarz later showed that if fj'(i) = 0 on an open interval (a, 6), then
F(x) is linear on the closed interval [a,b].