Problem Statement
Let $B_1$ be the Boolean algebra of sets of integers modulo sets of density $0$ (that is, in which two sets are equivalent if and only if they differ by a set of density $0$) and let $B_2$ be the Boolean algebra of sets modulo sets of logarithmic density $0$.
Prove that $B_1$ and $B_2$ are not isomorphic.
Prove that $B_1$ and $B_2$ are not isomorphic.
Categories:
Algebra
Progress
A question of Erdős and Ulam, who noted that the Boolean algebra modulo finite sets is not isomorphic to either of these two, because the former has no upper bound while the other two do.They thought they had a proof of the main claim, but this was 'lost'. Erdős [Er81b] writes 'When I first visited Ulam in 1943 or 1944 in Madison we had the proof, then six months later we had forgotten the proof, and had to reconstruct it, so it seems that the proof should have been correct. Now the proof is gone and nobody can prove it. This problem should be settled; perhaps I should offer a hundred dollars for a proof (or a disproof) that these two Boolean algebras are not isomorphic. If it is trivial I well deserve to have to pay the hundred dollars.'
In fact Just and Krawczyk [JuKr84] proved, assuming the continuum hypothesis, that these two algebras are isomorphic. It is unclear whether this counts as a solution - in [JuKr84] they claim Erdős and Ulam asked about this only under the continuum hypothesis, but this is not mentioned in [Er81b].
On the other hand, the question whether these two algebras are isomorphic appears in [VMR80] as Question 48, where it is attributed to Erdős, and there it is only asked about assuming the continuum hypothesis.
Source: erdosproblems.com/1123 | Last verified: January 19, 2026