Martin’s axiom and the continuum hypothesis

MA0 always holds

Given a countableMathworldPlanetmath collectionMathworldPlanetmath of dense subsets of a partial orderMathworldPlanetmath, we can selected a set pnn<ω such that pn is in the n-th dense subset, and pn+1pn for each n. Therefore CH implies MA.

If MAκ then 20>κ, and in fact 2κ=20

κ0, so 2κ20, hence it will suffice to find an surjective function from P(0) to P(κ).

Let A=Aαα<κ, a sequencePlanetmathPlanetmath of infinite subsets of ω such that for any αβ, AαAβ is finite.

Given any subset Sκ we will construct a function f:ω{0,1} such that a unique S can be recovered from each f. f will have the property that if iS then f(a)=0 for finitely many elements aAi, and if iS then f(a)=0 for infinitely many elements of Ai.

Let P be the partial order (under inclusion) such that each element pP satisfies:

  • p is a partial functionMathworldPlanetmath from ω to {0,1}

  • There exist i1,,inS such that for each j<n, Aijdom(p)

  • There is a finite subset of ω, wp, such that wp=dom(p)-j<nAij

  • For each j<n, p(a)=0 for finitely many elements of Aij

This satisfies ccc. To see this, consider any uncountable sequence S=pαα<ω1 of elements of P. There are only countably many finite subsets of ω, so there is some wω such that w=wp for uncountably many pS and pw is the same for each such element. Since each of these function’s domain covers only a finite number of the Aα, and is 1 on all but a finite number of elements in each, there are only a countable number of different combinationsMathworldPlanetmathPlanetmath available, and therefore two of them are compatible.

Consider the following groups of dense subsets:

  • Dn={pPndom(p)} for n<ω. This is obviously dense since any p not already in Dn can be extended to one which is by adding n,1

  • Dα={pPdom(p)Aα} for αS. This is dense since if pDα then p{a,1aAαdom(p)} is.

  • For each αS, n<ω, Dn,α={pPmnp(m)=0} for some m<ω. This is dense since if pDn,α then dom(p)Aα=Aα(wpjAij). But wp is finite, and the intersectionMathworldPlanetmathPlanetmath of Aα with any other Ai is finite, so this intersection is finite, and hence bounded by some m. Aα is infinite, so there is some mxAα. So p{x,0}Dn,α.

By MAκ, given any set of κ dense subsets of P, there is a generic G which intersects all of them. There are a total of 0+|S|+(κ-|S|)0=κ dense subsets in these three groups, and hence some generic G intersecting all of them. Since G is directed, g=G is a partial function from ω to {0,1}. Since for each n<ω, GDn is non-empty, ndom(g), so g is a total function. Since GDα for αS is non-empty, there is some element of G whose domain contains all of Aα and is 0 on a finite number of them, hence g(a)=0 for a finite number of aAα. Finally, since GDn,α for each n<ω, αS, the set of nAα such that g(n)=0 is unboundedPlanetmathPlanetmath, and hence infinite. So g is as promised, and 2κ=20.

Title Martin’s axiom and the continuum hypothesisMathworldPlanetmath
Canonical name MartinsAxiomAndTheContinuumHypothesis
Date of creation 2013-03-22 12:55:05
Last modified on 2013-03-22 12:55:05
Owner Henry (455)
Last modified by Henry (455)
Numerical id 4
Author Henry (455)
Entry type Result
Classification msc 03E50