Suslin's problem

In set theory, the Suslin hypothesis postulated ( named after the Russian mathematician Mikhail Yakovlevich Suslin ) a special characterization of the set of real numbers. It is neither provable nor refutable in the usual system of Zermelo -Fraenkel set theory.

Motivation

Georg Cantor showed the following order theory the real numbers: A linear order is isomorphic to if the following holds:

• Is unlimited: For each there is so.
• Is tight: For every pair with, there is a so.
• Is complete: every Dedekind cut of has a supremum in.
• Separable: contains a countable dense subset.

Any such linear order also meets the so-called countable antichain condition:

• Each family of open, pairwise disjoint intervals is at most countable.

The proof of this additional property directly follows the separability. Suslin set in 1920 hypothesized that the converse also holds, ie are separability and countable antichain condition is equivalent.

Formulation and consequences

The Suslin hypothesis can be expressed thus:

"Every unrestricted, dense, full linear order that satisfies the countable antichain condition is isomorphic to the order of the real numbers. "

Ronald Jensen showed in 1968 that in the model of constructible sets the Suslin hypothesis is false. Using the forcing method Robert M. Solovay and Stanley Tennenbaum constructed in 1971 a model in which the hypothesis is true, so it is neither provable nor refutable.