Suslin's Problem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A nonempty linear order $(L,<)$ is dense if between any two distinct points there is a third point, has no endpoints if every point has a smaller and a larger point, and is Dedekind complete if every nonempty subset bounded above has a least upper bound in $L$. Its countable chain condition means that every family of pairwise disjoint nonempty open intervals $(a,b)=\{x:a<x<b\}$ is countable. An order isomorphism is a bijection preserving the order in both directions.

Statement recorded in the supplied source.

Must every dense Dedekind-complete linear order without endpoints satisfying the countable chain condition be order-isomorphic to $(\mathbb R,<)$? [S2]

Short English statement

Do completeness, order density, lack of endpoints and countability of disjoint intervals characterize the ordered real line? This is the historical Suslin question; its answer is independent of ZFC, relative to the usual consistency assumption.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1176 (SET-002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Chris Lambie-Hanson and Assaf Rinot, A forcing axiom deciding the generalized Souslin hypothesis, arXiv:1708.06932v2 (2018). Section1.1, page1 and footnote1. https://arxiv.org/pdf/1708.06932.