## 580 -Some choiceless results (2)

January 25, 2009

There are a few additional remarks on the Schröder-Bernstein theorem worth mentioning. I will expand on some of them later, in the context of descriptive set theory.

The dual Schröder-Bernstein theorem (dual S-B) is the statement “Whenever $A,B$ are sets and there are surjections from $A$ onto $B$ and from $B$ onto $A,$ then there is a bijection between $A$ and $B$.”

* This follows from the axiom of choice. In fact, ${\sf AC}$ is equivalent to: Any surjective function admits a right inverse. So the dual S-B follows from choice and the S-B theorem.

* The proofs of S-B actually show that if one has injections $f:A\to B$ and $g:B\to A$, then one has a bijection $h:A\to B$ contained in $f\cup g^{-1}$. So the argument above gives the same strengthened version of the dual S-B. Actually, over ${\sf ZF}$, this strengthened version implies choice. This is in Bernhard Banaschewski, Gregory H. Moore, The dual Cantor-Bernstein theorem and the partition principle, Notre Dame J. Formal Logic 31 (3), (1990), 375-381.

* If $j : {}x \to y$ is onto, then there is $k:{\mathcal P}(y)\to {\mathcal P}(x)$ 1-1, so if there are surjections in both directions between $A$ and $B$, then ${\mathcal P}(A)$ and ${\mathcal P}(B)$ have the same size. Of course, this is possible even if $A$ and $B$ do not.

Open question. (${\sf ZF}$) Does the dual Schröder-Bernstein theorem imply the axiom of choice?

* The dual S-B is not a theorem of ${\sf ZF}$.

## 175 -A curious series

November 10, 2008

I just learned from the textbook that apparently whether the series

$\displaystyle \sum_{n=1}^\infty\frac1{n^3\sin^2 n}$

converges is still open, which I find rather surprising. The reference the book lists is the book Mazes for the Mind by Clifford Pickover, St. Martin Press, NY, from 1992, but Dr. Pickover has informed me that he believes the problem is still unresolved; he also discusses it in his book The Mathematics of Oz, Cambridge University Press, 2002. I would be very curious to hear from updates or suggestions, if you have any.

Here is a slightly technical (and very quick, and not particularly deep) observation: The issue seems to be to quantify how small $\sin^2 n$ is, when it is small, or more precisely, how sparse the set of values of $n$ is for which the sine function is “significantly small.” One could start by looking at $n$ so that $|n-m\pi|$ is small for some $m$, so we are led to consider the standard convergent approximations to $\pi$, satisfying $\displaystyle \left|\frac nm-\pi\right|<\frac1{m^2}$. This means that $1/(n^3\sin^2 n)$ is close to, but slightly larger than, $\displaystyle\frac{1/\pi^2}n,$ and so the question leads us to the problem of how sparse the sequence of numerators of the rational approximations to $\pi$ actually is, something about which I don’t know of any results.

Below I display some graphs for the partial sums of the series. Let $\displaystyle S(n)=\sum_{k=1}^n\frac1{k^3\sin^2 k}$. The first graph shows $n$ vs. $S(n)$ for $1\le n\le 100$. In the other graphs, $n$ goes up to 300, 1000, and 100000. (Thanks to Richard Ketchersid for the code.) It is not clear to me that the last graph is accurate or that it allows us to draw any conclusions (it certainly seems to suggest that the series converges to a number slightly larger than 30); it may well be that further jumps are beyond the range I chose, or that the approximations Maple uses in its computations are not fine enough to examine very large values of the series.