Cardinality Question: Riemann's Rearrangement Theorem
Fred Galvin
> Riemann's rearrangement theorem (Umordnungssatz) states that a
> conditionally convergent series of real numbers can be made to
> converge to any desired real number by just rearranging its
> positive and negative terms. Considering the set of all
> rearrangements of such a series it is clear that it has a
> cardinaliity at least equal to that of the reals, since every real
> corresponds to at least one of these rearrangements.
> Question: Is the cardinality of the rearrangements GREATER than
> that of the reals, and if so, what is it equal to?
No, the cardinality of the rearrangements is no greater than that of
the reals. In fact, the set of *all* infinite series/sequences of real
numbers has the same cardinality as the set of all real numbers. I
suppose you know that the set of real numbers has the same cardinality
as the set of all infinite sequences of 0's and 1's. Now, if we can
make a real number correspond to an infinite sequence of binary
digits, then an infinite sequence of real numbers corresponds to a
double sequence of binary digits. But there are no more double
sequences than simple infinite sequences, because NxN is equivalent to
N. In terms of cardinal arithmetic, abbreviating n = card(N) = aleph_0
and c = card(R) = 2^n, we have c^n = (2^n)^n = 2^(n*n) = 2^n = c.
--
"Any clod can have the facts, but having opinions is an art."--McCabe
mathc...@hotmail.com
Dave L. Renfro
[sci.math Mon, 18 Sep 2000 02:20:50 GMT]
<http://forum.swarthmore.edu/epigone/sci.math/snysmermbror>
wrote
As Fred Galvin has already pointed out, the cardinality of
all sequences of real numbers is c (= cardinality of the
set of real numbers).
However, sci.math readers might be interested in
a Baire category result involving Riemann's rearrangement
theorem, due to Ralph P. Agnew (1940) [1].
Let E be the set of permutations of positive integers with
the Frechet metric. Thus, if x and y are permutations, then
d(x,y) = SUM from n=1 to infinity of
2^(-n) * { |x(n) - y(n)| / [1 + |x(n) - y(n)|] }.
Although E is not complete, E has the property that each
neighborhood of each point is 2'nd category. Agnew credits
Mark Kac with this problem: Given any conditionally convergent
series a_1 + a_2 + ..., what is the category (in the sense of
Baire) of E - A, where A consists of those permutations x for
which a_x(1) + a_x(2) + ... converges?
Agnew proves that A is a first category set in E. In fact,
Agnew actually proves that the larger set B of permutations for
which the sum has unilaterally bounded partial sums is first
category in E.
In other words, given ANY conditionally convergent series,
almost all (in the Baire category sense) of its rearrangements
result in a series whose partial sums have these properties:
(a) the lim inf of the partial sum is -infinity
(b) the lim sup of the partial sum is +infinity
Sengupta (1950) [5] proves that B is F_sigma in E and that
B has cardinality c. In fact, Sengupta actually proves that
given any real numbers L < M, there are c permutations in E
which result in rearrangements of the given conditionally
convergent series whose partial sums are bounded between L and M.
Sengupta (1956) [6] proves that A is dense in E and that the map
f: A --> 'reals' defined by f(x) = 'the sum of the given series
with "x-rearrangement"' is everywhere discontinuous, open, and
not closed. In fact, as Sengupta says IN THE LAST SENTENCE OF
HIS PROOF that A is dense (but NOT in the statement of the
theorem itself), his proof actually shows that given any real
number L the set A(L) is dense in E, where A(L) is the set of
x in E such that f(x) = L. [This is an excellent example for
the comments I make near the end of my Sept. 18, 2000 sci.math
post at
<http://forum.swarthmore.edu/epigone/sci.math/jahendcrang>.]
Ganguli and Lahiri (1968) [2] extend this more precise result
by Sengupta involving the sets A(L). Let L and M be two extended
real numbers (i.e. the reals along with -infinity and +infinity)
such that L = M or L < M. We denote by A(L,M) the set of those
permutations x in A such that
(a) the lim inf of the "x-rearranged" partial sum is L
and
(b) the lim sup of the "x-rearranged" partial sum is M.
Note that A(L,L) = A(L) if L is a real number.
Ganguli and Lahiri prove that A(L,M) is dense in E and that
A(L,M) has cardinality c.
There are quite a number of papers that deal with variations
on these issues. One variation is to consider subseries of
a given conditionally convergent series. Another variation
involves taking a given divergent series SUM[a_n] of positive
terms such that a_n --> 0 and considering sums of the form
SUM[x(n)*a_n], where x is a mapping from the positive integers
into {-1, 1}. The set of all such x's under the Frechet
metric forms a complete metric space, and thus subsets of
it corresponding to various convergence behaviors of the
resulting series SUM[x(n)*a_n] can be investigated in the
same way that E and its subsets A, B, A(L), etc. above are.
Finally, one way to state Riemann's rearrangement theorem is
that given ANY series of real numbers, the set of all finite
sums of rearrangements of this series is either the empty set,
a singleton set, or all real numbers. For series of complex
numbers the set of all finite sums of rearrangements is either
the empty set, a singleton set, a line in the complex plane,
or the entire complex plane (Levy, 1905). More generally, for
series in R^n, the set of all finite sums of rearrangements
is either the empty set, a singleton, a translate of a proper
subspace, or all of R^n (Steinitz, 1913). The proofs of these
results are a bit more involved than the usual proof of
Riemann's rearrangement theorem. See McNeill (1997) [3] (which
contains a number of interesting historical remarks) and
Rosenthal (1987) [4].
[1] Ralph Palmer Agnew, "On rearrangements of series", Bull.
Amer. Math. Soc. 46 (1940), 797-799.
[2] P. L. Ganguli and B. K. Lahiri, "Some results on certain sets
of series", Czech. Math. J. 18 (1968), 589-594.
[3] Kerry Smith McNeill, "Rearrangement of series", Pi Mu Epsilon
Journal 10 (1997), 547-555.
[4] P. Rosenthal, "The remarkable theorem of Levy and Steinitz",
Amer. Math. Monthly 94 (1987), 342-351.
[5] H. M. Sengupta, "On rearrangements of series", Proc.
Amer. Math. Soc. 1 (1950), 71-75.
[6] H. M. Sengupta, "Rearrangements of series", Proc. Amer.
Math. Soc. 7 (1956), 347-350.
Dave L. Renfro
mathc...@hotmail.com
one should exhibit the method by which a doubly infinite sequence of
binary digits can be made to correspond to a singly infinite sequence
of binary digits. A rearrangement of the digits of the double sequence
by the Cauchy diagonal method offers such a method.
On Sun, 17 Sep 2000 23:54:00 -0500, Fred Galvin
<gal...@math.ukans.edu> wrote:
>On Mon, 18 Sep 2000 mathc...@hotmail.com wrote:
>