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Claim. Let $\mathcal{Z}_0$ be the ideal of sets of positive integers of asymptotic density $0$ and $\mathcal{Z}_{\log}$ the ideal of sets of logarithmic density $0$, and let $B_1=\mathcal{P}(\mathbb{N})/\mathcal{Z}_0$ and $B_2=\mathcal{P}(\mathbb{N})/\mathcal{Z}_{\log}$ be the quotient Boolean algebras of Problem 1123. Assuming the continuum hypothesis, $B_1$ and $B_2$ are isomorphic. So the non-isomorphism that the problem asks to prove fails in every model of CH, and ZFC does not prove it.

Covers. The unprovability half of the independence: ZFC does not prove $B_1\not\cong B_2$, because $B_1\cong B_2$ holds under CH. Whether ZFC refutes $B_1\not\cong B_2$ is settled the other way, under the Open Coloring Axiom and Martin's Axiom, on Farah's page, which carries the full claim that the statement is independent of ZFC.

Formulation. The site's commentary records a disagreement about what Erdős and Ulam asked. Just and Krawczyk say the question was posed under CH, and Question 48 of the list of van Douwen, Monk and Rubin [VMR80], attributed to Erdős, asks it only under CH; Erdős's own account in the Scottish Book [Er81b] states it with no such hypothesis and offers the prize for a proof or a disproof. Read under CH, this theorem answers the question: the algebras are isomorphic, the opposite of what Erdős asked to prove. Read in ZFC, as the catalog's statement is, it is one half of the independence. The site's label and this corpus's standing follow the ZFC reading; the CH reading is disclosed here and not given a page of its own.

Source. W. Just and A. Krawczyk, On certain Boolean algebras $\mathcal{P}(\omega)/I$, Trans. Amer. Math. Soc. 285 (1984), no. 1, 411--429, doi:10.1090/S0002-9947-1984-0748847-1. The issue is dated September 1984 and carries no day, so this page's date is the first of that month. The paper is not held by this corpus; the statement is recorded as the site's commentary states it. Nothing on this page is independently reviewed by this project.

Acceptance. Refereed: the result is a journal paper in the Transactions of the American Mathematical Society. Reviewed: the curator of erdosproblems.com, T. F. Bloom, labels Problem 1123 independent of ZFC and credits Just and Krawczyk with the isomorphism under the continuum hypothesis (problem page last edited 5 March 2026). The curator is independent of the authors.