Suslin's problem
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Suslin's problem in mathematics is a question about orders posed by M. Suslin in the early 1920s. It turned out much later that it cannot be answered: 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.
The original question was this. 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 intervals in R is countable (this is also known as the "countable chain condition", ccc)
is R necessarily order-isomorphic to the real line R?
Note that if the fourth condition above about collections of intervals is exchanged with
- there exists a countable dense subset in R (i.e., R is a separable space)
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.
The Suslin hypothesis is the assertion that there are no Suslin lines, that is every countable-chain-condition dense complete linear order without endpoints is isomorphic to the real line. Equivalently, it is the assertion that every infinite tree of height ω1 either has a branch of length ω1 or an antichain of cardinality ω1.
The generalized Suslin hypothesis asserts that for every infinite regular cardinal κ every tree of height κ either has a branch of length κ or an antichain of cardinality κ.
The Suslin hypothesis is independent of ZFC, and is independent of both the generalized continuum hypothesis and of the negation of the continuum hypothesis. However, Martin's axiom, when combined with the negation of the Continuum Hypothesis, implies the Suslin Hypothesis. It is not known whether the Generalized Suslin Hypothesis is consistent with the Generalized Continuum Hypothesis; however, since the combination implies the negation of the square principle at a singular strong limit cardinal - in fact, at all singular cardinals and all regular successor cardinals - it implies that the axiom of determinacy holds in L(R) and is believed to imply the existence of an inner model with a superstrong cardinal.
(Suslin is also sometimes written as Souslin.)