Editing Suslin's problem (section)

==Implications==
Any totally ordered set that is ''not'' isomorphic to '''R''' but satisfies properties 1–4 is known as a '''Suslin line'''. The '''Suslin hypothesis''' says that there are no Suslin lines: that every countable-chain-condition dense complete linear order without endpoints is isomorphic to the real line. An equivalent statement is that every [[tree (set theory)|tree]] of height ω<sub>1</sub> either has a branch of length ω<sub>1</sub> or an [[antichain]] of [[cardinality]] ℵ<sub>1</sub>.

The '''generalized Suslin hypothesis''' says 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 existence of Suslin lines is equivalent to the existence of [[Suslin tree]]s and to [[Suslin algebra]]s.

The Suslin hypothesis is independent of ZFC.
{{harvtxt|Jech|1967}} and {{harvtxt|Tennenbaum|1968}} independently used [[Forcing (mathematics)|forcing methods]] to construct models of ZFC in which Suslin lines exist. [[Ronald Jensen|Jensen]] later proved that Suslin lines exist if the [[diamond principle]], a consequence of the [[axiom of constructibility]] V&nbsp;=&nbsp;L, is assumed. (Jensen's result was a surprise, as it had previously been [[conjecture]]d that V&nbsp;=&nbsp;L implies that no Suslin lines exist, on the grounds that V&nbsp;=&nbsp;L implies that there are "few" sets.) On the other hand, {{harvtxt|Solovay|Tennenbaum|1971}} used forcing to construct a model of ZFC without Suslin lines; more precisely, they showed that [[Martin's axiom]] plus the negation of the continuum hypothesis implies the Suslin hypothesis.

The Suslin hypothesis is also independent of both the [[generalized continuum hypothesis]] (proved by [[Ronald Jensen]]) and of the negation of the [[continuum 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 cardinal]]s and all regular [[successor cardinal]]s—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]].