Problem 1123

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.

Source: erdosproblems.com/1123

Set theory

Wiki page

Status

SolvedIndependent

An accepted solution exists. The statement can be neither proved nor disproved from the standard axioms of set theory.

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.

Problem 1123

Status

Proof claims

Discussion