Suslin's problem
Encyclopedia
In mathematics
, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin
in a work published posthumously in 1920.
It has been shown to be independent
of the standard axiomatic system of set theory
known as ZFC: the statement can neither be proven nor disproven from those axioms.
(Suslin is also sometimes written with the French transliteration as Souslin, from the Cyrillic Суслин.)
is R necessarily 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 isomorphic to R.
s. Suslin lines exist if the additional constructibility axiom V equals L
is assumed.
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
of height ω1 either has a branch of length ω1 or an antichain
of cardinality
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 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin
Mikhail Yakovlevich Suslin
Mikhail Yakovlevich Suslin was a Russian mathematician who made major contributions to the fields of general topology and descriptive set theory....
in a work published posthumously in 1920.
It has been shown to be independent
Independence (mathematical logic)
In mathematical logic, independence refers to the unprovability of a sentence from other sentences.A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that...
of the standard axiomatic system of set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
known as ZFC: the statement can neither be proven nor disproven from those axioms.
(Suslin is also sometimes written with the French transliteration as Souslin, from the Cyrillic Суслин.)
Formulation
Given a non-empty totally ordered set R with the following four properties:- R does not have a least nor a greatest elementGreatest elementIn mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...
- the order on R is denseDense orderIn mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x In mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x...
(between any two elements there is another) - the order on R is completeCompleteness (order theory)In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set . A special use of the term refers to complete partial orders or complete lattices...
, in the sense that every non-empty bounded subset has a supremumSupremumIn mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
and an infimumInfimumIn mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used... - every collection of mutually disjoint non-empty open intervalInterval (mathematics)In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
s in R is countable (this is the countable chain conditionCountable chain conditionIn order theory, a partially ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable. There are really two conditions: the upwards and downwards countable chain conditions. These are not equivalent...
, ccc)
is R necessarily order-isomorphic
Order isomorphism
In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets . Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that one of...
to the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
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 treeSuslin tree
In mathematics, a Suslin tree is a tree of height ω1 such thatevery branch and every antichain is at most countable. Every Suslin tree is an Aronszajn tree....
s. Suslin lines exist if the additional constructibility axiom V equals L
Axiom of constructibility
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as V = L, where V and L denote the von Neumann universe and the constructible universe, respectively.- Implications :The axiom of...
is assumed.
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
Tree (set theory)
In set theory, a tree is a partially ordered set In set theory, a tree is a partially ordered set (poset) In set theory, a tree is a partially ordered set (poset) (T, In set theory, a tree is a partially ordered set (poset) (T, ...
of height ω1 either has a branch of length ω1 or an antichain
Antichain
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two elements in the subset are incomparable. Let S be a partially ordered set...
of cardinality
The generalized Suslin hypothesis says that for every infinite regular cardinal
Regular cardinal
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. So, crudely speaking, a regular cardinal is one which cannot be broken into a smaller collection of smaller parts....
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
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
. However, Martin's axiom
Martin's axiom
In the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...
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
Limit cardinal
In mathematics, limit cardinals are certain cardinal numbers. A cardinal number λ is a weak limit cardinal if λ is neither a successor cardinal nor zero. This means that one cannot "reach" λ by repeated successor operations...
—in fact, at all singular cardinals and all regular successor cardinals—it implies that the axiom of determinacy
Axiom of determinacy
The axiom of determinacy is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person games of length ω with perfect information...
holds in L(R) and is believed to imply the existence of an inner model
Inner model
In mathematical logic, suppose T is a theory in the languageL = \langle \in \rangleof set theory.If M is a model of L describing a set theory and N is a class of M such that \langle N, \in_M, \ldots \rangle...
with a superstrong cardinal.