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The following problem was in The American Mathematical Monthly :

A generalized rectangle is $E \times F$ for any subsets $E,F$ of $\Bbb R$ (the reals). If $\mathscr{B}$ is the smallest countably closed Boolean algebra of subsets of $\Bbb R^2$ such that $\mathscr{B}$ contains every generalized rectangle, does $\mathscr{B}$ contain every subset of $\Bbb R^2$?

My attempt was to let $<^*$ be a well-order on $\Bbb R$, isomorphic to the cardinal ordinal $2^\omega$, and try to show that $ C= \{ (x,y) : x <^* y \} \not \in\mathscr{B}$ but I got stuck.

So is it true that $C \not \in \mathscr{B} , assuming the Continuum Hypothesis? ................The reason I have added CH is explained in my comment in response to another comment. The AMM question is undecidable in ZFC.... I dk whether my Q can be answered by elementary means.

DanielWainfleet
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  • Which issue of the AMM was this in? – user642796 Aug 22 '15 at 03:54
  • This question was the first part of Problem 99 (due to S. Ulam) in The Scottish Book: "By a product set in the unit square, we understand the set of all pairs $(x,y)$ where $x$ belongs to a given set $A$, $y$ to a given set $B$. Do there exist sets which cannot be obtained through the operations of forming countable sums and differences of sets starting from product sets? Do there exist nonprojective sets with respect to product subsets?" – bof Aug 22 '15 at 04:44
  • It may be helpful to work under the assumption that $2^\omega=\omega_1$. – bof Aug 22 '15 at 04:46
  • See http://math.stackexchange.com/questions/545640/does-mathcal-p-mathbb-r-otimes-mathcal-p-mathbb-r-mathcal-p/546033#546033 – hot_queen Aug 22 '15 at 04:48
  • I have put an addendum on,and modified the Q. In a comment to a link to the link in hot_queen's comment, there is a reference to a proof that the AMM question's answer is "yes" if CH (continuum hypothesis) but that is undecidable in ZFC. Some Q's in AMM are asterisked,which means neither the editors nor the proposer had a solution. – DanielWainfleet Aug 22 '15 at 17:48
  • For much more on this, see MO question http://mathoverflow.net/a/81491/454 – GEdgar Aug 22 '15 at 18:07

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