a shorter proof: Martin’s axiom and the continuum hypothesis

This is another, shorter, proof for the fact that MA0 always holds.

Let (P,) be a partially ordered setMathworldPlanetmath and 𝒟 be a collectionMathworldPlanetmath of subsets of P. We remember that a filter G on (P,) is 𝒟-genericPlanetmathPlanetmathPlanetmath if GD for all D𝒟 which are dense in (P,). (In this context “dense” means: If D is dense in (P,), then for every pP there’s a dD such that dp.)

Let (P,) be a partially ordered set and 𝒟 a countableMathworldPlanetmath collection of dense subsets of P. Then there exists a 𝒟-generic filter G on P. Moreover, it could be shown that for every pP there’s such a 𝒟-generic filter G with pG.


Let D1,,Dn, be the dense subsets in 𝒟. Furthermore let p0=p. Now we can choose for every 1n<ω an element pnP such that pnpn-1 and pnDn. If we now consider the set G:={qPn<ω s.t. pnq}, then it is easy to check that G is a 𝒟-generic filter on P and pG obviously. This completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof. ∎

Title a shorter proof: Martin’s axiom and the continuum hypothesisMathworldPlanetmath
Canonical name AShorterProofMartinsAxiomAndTheContinuumHypothesis
Date of creation 2013-03-22 13:53:58
Last modified on 2013-03-22 13:53:58
Owner x_bas (2940)
Last modified by x_bas (2940)
Numerical id 11
Author x_bas (2940)
Entry type Proof
Classification msc 03E50
Defines 𝒟-generic
Defines generic
Defines dense