Problem 1123

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claims/: The 2 claim pages of Problem 1123, one per claimant's result; the problem's standing derives from them.

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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.

Status. Independent.

Source. erdosproblems.com/1123, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1123, https://www.erdosproblems.com/1123.

References.

• [ARS85] Abraham, Uri and Rubin, Matatyahu and Shelah, Saharon, On the consistency of some partition theorems for continuous colorings, and the structure of $\aleph_1$-dense real order types. Ann. Pure Appl. Logic (1985), 123-206.
• [Er81b] Erdős, P., My Scottish Book 'Problems'. The Scottish Book (1981), 27-35 (page numbers are given for the 2nd edition of The Scottish Book).
• [Fa00] Farah, Ilijas, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers. Mem. Amer. Math. Soc. (2000), xvi+177.
• [JuKr84] Just, Winfried and Krawczyk, Adam, On certain Boolean algebras ${\scr P}(\omega )/I$. Trans. Amer. Math. Soc. (1984), 411-429.
• [VMR80] van Douwen, Eric K. and Monk, J. Donald and Rubin, Matatyahu, Some questions about Boolean algebras. Algebra Universalis (1980), 220-243.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• abraham_1985_consistency_partition_theorems_continuous_colorings_structure
• abraham_1985_consistency_partition_theorems_continuous_colorings_structure / theorem_3_1