Suslin's problem

Suslin's problem in mathematics is the following question posed by M. Suslin in the early 1920s: given a non-empty totally ordered set R with the following four properties

1. R does not have a smallest nor a largest element
2. the order on R is dense (between any two elements there's another one)
3. the order on R is complete, in the sense that every non-empty bounded set has a supremum and an infimum
4. any collection of mutually disjoint non-empty open intervalss in R is countable (this is also known as the "countable chain condition", ccc)

is R necessarily order-isomorphic to the real line R?

In the 1960s, it was proved that the question is undecidable from the standard axiomatic system of set theory known as ZFC: the statement can neither be proven nor disproven from those axioms.

Note that if the fourth condition above about collections of intervals is exchanged with

• there exists a countable dense subset in R

then the answer is indeed yes: any such set R is necessarily isomorphic to R.

Any totally ordered set that is not isomorphic to R but satisfies 1) - 4) is known as a Suslin line. The existence of Suslin lines has been proven to be equivalent to the existence of Suslin trees. Suslin lines exist if the additional constructibility axiom V equals L is assumed.