"Suppose L The real line is a the only endless dense complete linear order with the LUB property, such that in which every family of pairwise disjoint intervals in L is countable. Then L is order isomorphic to the real numbers.countable."
Of course, if you replace the anti-chain condition with
This statement generalizes the property familar characterization of being separable the real line (due to Cantor) as the unique endless dense complete linear order having a countable dense subset), then the statement is trueset. The property weakens Souslin inquired whether this separability condition can be weaked to the requirement only that all condition on families of disjoint intervalsare countable. (Here, complete means that the order as the LUB property.)
This statement is known as Souslin's Hypothesis, and it is independent of ZFC. It is false under the combinatorical assertion known as Diamond, but follows from Martin's Axiom at Aleph_1. The proof that the statement is consistent, due to Solovay and Tennenbaum, is highly important in the history of set theory, since it required the development of iterated forcing, now a fundamental tool.
