Suslin hypothesis

In set theory , the Suslin hypothesis (named after the Russian mathematician Michail Jakowlewitsch Suslin ) postulates a special characterization of the set of real numbers . In the usual system of Zermelo-Fraenkel set theory, it can neither be proven nor refuted.

motivation

Georg Cantor showed the following order theory characterization of the real numbers: a linear order is if and isomorphic to if the following applies: ${\displaystyle \langle P,<\rangle }$ ${\displaystyle \langle \mathbb {R} ,<\rangle }$

• ${\displaystyle P}$is unlimited: there is such a thing for everyone . ${\displaystyle p\in P}$ ${\displaystyle q,r\in P}$ ${\displaystyle q<p<r}$
• ${\displaystyle P}$is tight : for every couple with there is one so . ${\displaystyle p,q\in P}$ ${\displaystyle p<q}$ ${\displaystyle r\in P}$ ${\displaystyle p<r<q}$
• ${\displaystyle P}$is complete : every Dedekindian pattern of has a supremum in . ${\displaystyle P}$ ${\displaystyle P}$
• ${\displaystyle P}$is separable : contains a countable, dense subset. ${\displaystyle P}$

Every such linear order also fulfills the so-called countable anti - chain condition : ${\displaystyle \langle P,<\rangle }$

• Every family of open , pairwise disjoint intervals of is at most countable. ${\displaystyle P}$

The proof of this additional property follows directly from the separability. In 1920, Suslin hypothesized that the reverse is also true, that is, separability and countable anti-chain conditions are equivalent.

Formulation and consequences

The Suslin hypothesis can thus be expressed:

Every unbounded, dense, complete linear order that satisfies the countable anti-chain condition is isomorphic to the order of the real numbers.

Ronald Jensen showed in 1968 that the Suslin hypothesis is wrong in the model of constructible sets . Using the forcing method, Robert M. Solovay and Stanley Tennenbaum constructed a model in 1971 in which the hypothesis is true , i.e. it is neither provable nor refutable. ${\displaystyle L}$

Individual evidence

1. ↑ Michail J. Suslin: Problème 3. In: Fundamenta Mathematicae. Volume 1. 1920, p. 223.
2. ↑ Ronald Jensen: Souslin's hypothesis is incompatible with V = L. In: Notices of the American Mathematical Society. Volume 15, 1968, p. 935.
3. ^ Robert M. Solovay, Stanley Tennenbaum: Iterated Cohen extensions and Souslin's problem. In: Annals of Mathematics. Series 2, Volume 94, 1971, pp. 201-245.

literature

• Jech, Thomas: Set Theory , Springer-Verlag Berlin Heidelberg (2006), ISBN 3-540-44085-2 .
• Kunen, Keneth: Set Theory: An Introduction to Independence Proofs , North-Holland (1980), ISBN 0-444-85401-0 .