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
- 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)
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.