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| In [[mathematics]], '''Suslin's problem''' is a question about [[totally ordered set]]s posed by [[Mikhail Yakovlevich Suslin]] in a work published posthumously in 1920.<ref>{{cite journal
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| |title=Problème 3
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| |last= Souslin
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| |first=M.
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| |journal=Fundamenta Mathematicae
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| |volume=1
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| |year=1920
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| |page=223
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| }}</ref>
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| It has been shown to be [[independence (mathematical logic)|independent]] of the standard axiomatic system of [[set theory]] known as [[ZFC]]: the statement can neither be proven nor disproven from those axioms.<ref>{{cite journal
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| |title=Iterated Cohen extensions and Souslin's problem
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| |last=Solovay
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| |first=R. M.
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| |coauthors=Tennenbaum, S.
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| |journal=Ann. Of Math. (2)
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| |volume=94
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| |year=1971
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| |pages=201–245
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| |doi=10.2307/1970860
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| |issue=2
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| |publisher=Annals of Mathematics
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| |jstor=1970860
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| }}</ref>
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| (Suslin is also sometimes written with the French transliteration as Souslin, from the Cyrillic Суслин.)
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| ==Formulation==
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| Given a [[non-empty]] [[totally ordered set]] ''R'' with the following four properties:
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| # ''R'' does not have a [[greatest element|least nor a greatest element]];
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| # the order on ''R'' is [[dense order|dense]] (between any two elements there is another);
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| # the order on ''R'' is [[completeness (order theory)|complete]], in the sense that every non-empty bounded subset has a [[supremum]] and an [[infimum]];
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| # every collection of mutually [[disjoint sets|disjoint]] non-empty open [[interval (mathematics)|interval]]s in ''R'' is [[countable]] (this is the [[countable chain condition]]).
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| Is ''R'' necessarily [[order isomorphism|order-isomorphic]] to the [[real line]] '''R'''?
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| If the requirement for the countable chain condition is replaced with the requirement that ''R'' contains a countable dense subset (i.e., ''R'' is a [[separable space]]) then the answer is indeed yes: any such set ''R'' is necessarily isomorphic to '''R'''.
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| ==Implications==
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| 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 tree]]s. Suslin lines exist if the additional constructibility axiom [[Axiom of constructibility|V equals L]] is assumed.
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| 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. Equivalently, 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 <math>\aleph_1.</math>
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| 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.
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| 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]] plus 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]].
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| == See also ==
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| * [[List of statements undecidable in ZFC]]
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| * [[AD+]]
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| ==Notes==
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| <references/>
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| ==References==
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| *{{springer|id=S/s091460
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| |first=V.N. |last=Grishin|title=Suslin hypothesis}}
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| {{Set theory}}
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| [[Category:Independence results]]
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| [[Category:Order theory]]
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