Editing Suslin's problem

In [[mathematics]], '''Suslin's problem''' is a question about [[totally ordered set]]s posed by {{harvs|txt|authorlink=Mikhail Yakovlevich Suslin|first=Mikhail Yakovlevich |last=Suslin|year=1920}} and published posthumously.
It has been shown to be [[independence (mathematical logic)|independent]] of the standard [[axiomatic system]] of [[set theory]] known as [[ZFC]]; {{harvtxt|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 as {{lang|fr|Souslin}}, from the Cyrillic {{lang|ru|Суслин}}.)
{{quotebox|right|width=50%
|quote={{lang|fr|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?
|source=The original statement of Suslin's problem from {{harv|Suslin|1920}}
}}

==Formulation==
Suslin's problem asks: Given a [[non-empty]] [[totally ordered set]] ''R'' with the four properties
# ''R'' does not have a [[greatest element and least element|least nor a greatest element]];
# the order on ''R'' is [[dense order|dense]] (between any two distinct elements there is another);
# the order on ''R'' is [[completeness (order theory)|complete]], in the sense that every non-empty bounded subset has a [[supremum]] and an [[infimum]]; and
# every collection of mutually [[disjoint sets|disjoint]] non-empty [[open interval]]s in ''R'' is [[countable]] (this is the [[countable chain condition]] for the [[order topology]] of ''R''),
is ''R'' necessarily [[order isomorphism|order-isomorphic]] to the [[real 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 order-isomorphic to '''R''' (proved by [[Georg Cantor|Cantor]]).

The condition for a [[topological space]] that every collection of non-empty disjoint [[open set]]s is at most countable is called the '''Suslin property'''.

==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]].

==See also==
* [[List of statements independent of ZFC]]
* [[Continuum hypothesis]]
* [[AD+|AD<sup>+</sup>]]
* [[Cantor's isomorphism theorem]]

==References==
* K. Devlin and H. Johnsbråten, The Souslin Problem, Lecture Notes in Mathematics (405) Springer 1974.
* {{citation|mr=0215729
|last=Jech|first= Tomáš
|title=Non-provability of Souslin's hypothesis
|journal=Comment. Math. Univ. Carolinae |volume=8 |year=1967 |pages=291–305}}
* {{citation
|title=Problème 3
|last= Souslin
|first=M.
|journal=Fundamenta Mathematicae
|volume=1
|year=1920
|page=223
|doi= 10.4064/fm-1-1-223-224
|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm1/fm1125.pdf
|doi-access=free
|ref={{harvid|Suslin|1920}}
}}
* {{citation
|last1=Solovay
|first1=R. M.
|last2=Tennenbaum
|first2=S.
|title=Iterated Cohen Extensions and Souslin's Problem
|journal=Annals of Mathematics
|date=1971
|volume=94
|issue=2
|pages=201–245
|doi=10.2307/1970860
|jstor=1970860}}
* {{citation|mr=0224456
|last=Tennenbaum
|first= S.
|title=Souslin's problem.
|journal=Proc. Natl. Acad. Sci. U.S.A.
|volume= 59
|year=1968
|issue=1
|pages= 60–63
|doi=10.1073/pnas.59.1.60
|pmc=286001
|pmid=16591594|bibcode=1968PNAS...59...60T
|doi-access=free
}}
* {{springer
|id=S/s091460
|first=V. N.
|last=Grishin
|title=Suslin hypothesis}}

{{Set theory}}

[[Category:Independence results]]
[[Category:Order theory]]