Martin’s axiom and the continuum hypothesis

$M{A}_{{\mathrm{\aleph }}_0}$ always holds

Given a countable collection of dense subsets of a partial order, we can selected a set such that $p_n$ is in the $n$-th dense subset, and $p_{n+1}\le p_n$ for each $n$. Therefore $CH$ implies $MA$.

If $M{A}_{\kappa }$ then $2^{{\mathrm{\aleph }}_0}>\kappa$, and in fact $2^{\kappa }=2^{{\mathrm{\aleph }}_0}$

$\kappa \ge {\mathrm{\aleph }}_0$, so $2^{\kappa }\ge 2^{{\mathrm{\aleph }}_0}$, hence it will suffice to find an surjective function from $\mathrm{P}({\mathrm{\aleph }}_0)$ to $\mathrm{P}(\kappa )$.

Let , a sequence of infinite subsets of $\omega$ such that for any $\alpha \ne \beta$, $A_{\alpha }\cap A_{\beta }$ is finite.

Given any subset $S\subseteq \kappa$ we will construct a function $f:\omega \to \{0,1\}$ such that a unique $S$ can be recovered from each $f$. $f$ will have the property that if $i\in S$ then $f(a)=0$ for finitely many elements $a\in A_i$, and if $i\notin S$ then $f(a)=0$ for infinitely many elements of $A_i$.

Let $P$ be the partial order (under inclusion) such that each element $p\in P$ satisfies:

• $p$ is a partial function from $\omega$ to $\{0,1\}$

• There exist $i_1,\mathrm{\dots },i_n\in S$ such that for each , $A_{i_j}\subseteq \mathrm{dom}(p)$

• There is a finite subset of $\omega$, $w_p$, such that

• For each , $p(a)=0$ for finitely many elements of $A_{i_j}$

This satisfies ccc. To see this, consider any uncountable sequence of elements of $P$. There are only countably many finite subsets of $\omega$, so there is some $w\subseteq \omega$ such that $w=w_p$ for uncountably many $p\in S$ and $p\upharpoonright w$ is the same for each such element. Since each of these function’s domain covers only a finite number of the $A_{\alpha }$, and is $1$ on all but a finite number of elements in each, there are only a countable number of different combinations available, and therefore two of them are compatible.

Consider the following groups of dense subsets:

• $D_n=\{p\in P\mid n\in \mathrm{dom}(p)\}$ for . This is obviously dense since any $p$ not already in $D_n$ can be extended to one which is by adding $⟨n,1⟩$

• $D_{\alpha }=\{p\in P\mid \mathrm{dom}(p)\supseteq A_{\alpha }\}$ for $\alpha \in S$. This is dense since if $p\notin D_{\alpha }$ then $p\cup \{⟨a,1⟩\mid a\in A_{\alpha }\setminus \mathrm{dom}(p)\}$ is.

• For each $\alpha \notin S$, , $D_{n,\alpha }=\{p\in P\mid m\ge n\wedge p(m)=0\}$ for some . This is dense since if $p\notin D_{n,\alpha }$ then $\mathrm{dom}(p)\cap A_{\alpha }=A_{\alpha }\cap \left(w_p\cup {\bigcup }_j{A}_{i_j}\right)$. But $w_p$ is finite, and the intersection of $A_{\alpha }$ with any other $A_i$ is finite, so this intersection is finite, and hence bounded by some $m$. $A_{\alpha }$ is infinite, so there is some $m\le x\in A_{\alpha }$. So $p\cup \{⟨x,0⟩\}\in D_{n,\alpha }$.

By $M{A}_{\kappa }$, given any set of $\kappa$ dense subsets of $P$, there is a generic $G$ which intersects all of them. There are a total of ${\mathrm{\aleph }}_0+|S|+(\kappa -|S|)\cdot {\mathrm{\aleph }}_0=\kappa$ dense subsets in these three groups, and hence some generic $G$ intersecting all of them. Since $G$ is directed, $g=\bigcup G$ is a partial function from $\omega$ to $\{0,1\}$. Since for each , $G\cap D_n$ is non-empty, $n\in \mathrm{dom}(g)$, so $g$ is a total function. Since $G\cap D_{\alpha }$ for $\alpha \in S$ is non-empty, there is some element of $G$ whose domain contains all of $A_{\alpha }$ and is $0$ on a finite number of them, hence $g(a)=0$ for a finite number of $a\in A_{\alpha }$. Finally, since $G\cap D_{n,\alpha }$ for each , $\alpha \notin S$, the set of $n\in A_{\alpha }$ such that $g(n)=0$ is unbounded, and hence infinite. So $g$ is as promised, and $2^{\kappa }=2^{{\mathrm{\aleph }}_0}$.

Title	Martin’s axiom and the continuum hypothesis
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