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.

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