Inmathematics,Suslin's problem is a question abouttotally ordered sets posed byMikhail Yakovlevich Suslin (1920) and published posthumously.It has been shown to beindependent of the standardaxiomatic system ofset theory known asZFC;Solovay & Tennenbaum (1971) showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent.
(Suslin is also sometimes written with the French transliteration asSouslin, from the CyrillicСуслин.)
Un ensemble ordonné (linéairement) sans sauts ni lacunes et tel que tout ensemble de ses intervalles (contenant plus qu'un élément) n'empiétant pas les uns sur les autres est au plus dénumerable, est-il nécessairement un continue linéaire (ordinaire)?
Is a (linearly) ordered set without jumps or gaps and such that every set of its intervals (containing more than one element) not overlapping each other is at most denumerable, necessarily an (ordinary) linear continuum?
Suslin's problem asks: Given anon-emptytotally ordered setR with the four properties
isR necessarilyorder-isomorphic to thereal lineR?
If the requirement for the countable chain condition is replaced with the requirement thatR contains a countable dense subset (i.e.,R is aseparable space), then the answer is indeed yes: any such setR is necessarily order-isomorphic toR (proved byCantor).
The condition for atopological space that every collection of non-empty disjointopen sets is at most countable is called theSuslin property.
Any totally ordered set that isnot isomorphic toR but satisfies properties 1–4 is known as aSuslin line. TheSuslin 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 everytree of height ω1 either has a branch of length ω1 or anantichain ofcardinality ℵ1.
Thegeneralized Suslin hypothesis says that for every infiniteregular cardinalκ every tree of heightκ either has a branch of lengthκ or an antichain of cardinalityκ. The existence of Suslin lines is equivalent to the existence ofSuslin trees and toSuslin algebras.
The Suslin hypothesis is independent of ZFC.Jech (1967) andTennenbaum (1968) independently usedforcing methods to construct models of ZFC in which Suslin lines exist.Jensen later proved that Suslin lines exist if thediamond principle, a consequence of theaxiom of constructibility V = L, is assumed. (Jensen's result was a surprise, as it had previously beenconjectured that V = L implies that no Suslin lines exist, on the grounds that V = L implies that there are "few" sets.) On the other hand,Solovay & Tennenbaum (1971) used forcing to construct a model of ZFC without Suslin lines; more precisely, they showed thatMartin's axiom plus the negation of the continuum hypothesis implies the Suslin hypothesis.
The Suslin hypothesis is also independent of both thegeneralized continuum hypothesis (proved byRonald Jensen) and of the negation of thecontinuum 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 thesquare principle at a singular stronglimit cardinal—in fact, at allsingular cardinals and all regularsuccessor cardinals—it implies that theaxiom of determinacy holds in L(R) and is believed to imply the existence of aninner model with asuperstrong cardinal.
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