---

Claim. Let $B_1=\mathcal{P}(\mathbb{N})/\mathcal{Z}_0$ and $B_2=\mathcal{P}(\mathbb{N})/\mathcal{Z}_{\log}$ be the quotients of the power set of the positive integers by the ideals of density zero and of logarithmic density zero, the algebras of Problem 1123. Assuming the Open Coloring Axiom (OCA) and Martin's Axiom (MA), $B_1$ and $B_2$ are not isomorphic (Corollary 3.4.4, as the site's commentary cites it). The axiom of Farah's lifting theory is Todorcevic's Open Coloring Axiom: for every coloring of the pairs of a separable metric space into an open color and its complement, either some uncountable set is homogeneous in the open color or the space is a countable union of sets homogeneous in the other. Every model of ZFC has a forcing extension in which MA and this axiom both hold. Farah states this for MA with his stronger axiom OCA$_\infty$ (arXiv:0705.3085v2, Section 2.4), citing his Cauchy nets and open colorings, Publ. Inst. Math. (Beograd) (N.S.) 64(78) (1998), 146--152, which proves the consistency of that axiom the way Todorcevic proved the consistency of OCA, with no large cardinal. So ZFC, if consistent, does not refute $B_1\not\cong B_2$. Both axioms also follow from the Proper Forcing Axiom. Under the continuum hypothesis the two algebras are isomorphic, by Just and Krawczyk, so ZFC does not prove $B_1\not\cong B_2$ either. The statement Erdős and Ulam asked to prove is therefore independent of ZFC, which is the site's label. This page carries the full claim: the memoir supplies the half that was missing after 1984, and the independence is the two halves together.

Depends on. Just and Krawczyk 1984 for the isomorphism under CH, the unprovability half.

History. Erdős and Ulam asked the question in the 1940s and believed they had a proof of the non-isomorphism, which was lost; Erdős's Scottish Book account [Er81b] offers a prize for a proof or a disproof. They already knew that the algebra of sets modulo finite sets is isomorphic to neither of the two. Whether the question was meant under CH is disputed, as the page of Just and Krawczyk records; under that reading their theorem answers it, and the catalog's ZFC reading is the one whose standing is recorded here.

Source. Ilijas Farah, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Mem. Amer. Math. Soc. 148 (2000), no. 702, xvi+177 pp., doi:10.1090/memo/0702. The publisher's record gives only the year, so this page's date is the first day of 2000. The memoir is not held by this corpus; its third chapter treats homomorphisms of analytic quotients under OCA, and the corollary is cited as the site cites it. The site's commentary credits U. Abraham, M. Rubin and S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of $\aleph_1$-dense real order types, Ann. Pure Appl. Logic 29 (1985), 123--206 (on the corpus's source card), with a model in which OCA and MA both hold. The axiom those authors call OCA, in which both colors are open, is a different axiom from Todorcevic's: Farah says so in his later paper on liftings, All automorphisms of the Calkin algebra are inner (arXiv:0705.3085v2, Section 2.3), which states Todorcevic's axiom, cites S. Todorcevic, Partition problems in topology, Contemp. Math. 84 (1989) for it and for the fact that PFA implies it, refers to the second chapter of the memoir for its reformulations, and names the variant with both colors open as the one of Abraham, Rubin and Shelah, not used there. So the consistency of the hypotheses of Corollary 3.4.4 relative to ZFC is recorded above through Farah's forcing, and the Abraham--Rubin--Shelah card is not cited as its source. Nothing on this page is independently reviewed by this project.

Acceptance. Refereed: the result is a memoir in the Memoirs of the American Mathematical Society. Reviewed: the curator of erdosproblems.com, T. F. Bloom, labels Problem 1123 independent of ZFC and credits Farah's corollary, with the consistency result of Abraham, Rubin and Shelah, for the half that completes the independence (problem page last edited 5 March 2026). The curator is independent of the author.