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