Martin's axiom
Martin's axiom in set theory a statement that the system is neither provable nor refutable in the usual Zermelo - Fraenkel. It was introduced in 1970 by Donald A. Martin and Robert M. Solovay.
Motivation
Let be a partially ordered set and a set of dense subsets of. Wanted is a filter that meets all the elements, ie not empty cuts ( then - called generic filter). The lemma of Rasiowa - Sikorski says that it is always possible for countable to find such. For überabzählbares the situation is different: if
- Or
- Uncountable antichains possesses,
There are generally no - generic filter.
Formulation
Martin's Axiom () is the statement
" For a partially ordered set that has only countable antichains, and a lot of dense subsets, there is a - generic filter. "
Does the continuum hypothesis, then every countably with necessary therefore holds trivially. However, it can also be models of construct, in which the continuum hypothesis does not apply.