- Suslin's problem
In
mathematics , Suslin's problem is a question abouttotally ordered set s posed byMikhail Yakovlevich Suslin in the early 1920s. [cite journal
title=Problème 3
last= Souslin
first=M.
journal=Fundamenta Mathematicae
volume=1
date=1920
pages=223] It has been shown to be independent of the standard axiomatic system ofset theory known asZFC : the statement can neither be proven nor disproven from those axioms. [cite journal
title=Iterated Cohen extensions and Souslin's problem
last=Solovay
first=R. M.
coauthors=Tennenbaum, S.
journal=Ann. of Math. (2)
volume=94
date=1971
pages=201–245
doi=10.2307/1970860](Suslin is also sometimes transliterated as Souslin, from the Cyrilic Суслин.)
Formulation
Given a
non-empty totally ordered set "R" with the following four properties:
# "R" does not have a least nor a greatest 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 subset has asupremum and aninfimum
# every collection of mutuallydisjoint non-empty open intervals in "R" iscountable (this is thecountable chain condition , ccc)is "R" necessarily order-isomorphic to thereal line R?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.Implications
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 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 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 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 thesquare principle at a singular stronglimit cardinal —in fact, at all singular cardinals and all regular successor cardinals—it implies that theaxiom of determinacy holds in L(R) and is believed to imply the existence of aninner model with asuperstrong cardinal .References
*springer|id=S/s091460
first=V.N. |last=Grishin|title=Suslin hypothesisSee also
*
List of statements undecidable in ZFC
*AD+
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