# Martin’s axiom and the continuum hypothesis

## $MA_{\aleph_{0}}$ always holds

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

## If $MA_{\kappa}$ then $2^{\aleph_{0}}>\kappa$, and in fact $2^{\kappa}=2^{\aleph_{0}}$

$\kappa\geq\aleph_{0}$, so $2^{\kappa}\geq 2^{\aleph_{0}}$, hence it will suffice to find an surjective function from $\operatorname{P}(\aleph_{0})$ to $\operatorname{P}(\kappa)$.

Let $A=\langle A_{\alpha}\rangle_{\alpha<\kappa}$, a sequence of infinite subsets of $\omega$ such that for any $\alpha\neq\beta$, $A_{\alpha}\cap A_{\beta}$ is finite.

Given any subset $S\subseteq\kappa$ we will construct a function $f:\omega\rightarrow\{0,1\}$ 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 $i_{1},\ldots,i_{n}\in S$ such that for each $j, $A_{i_{j}}\subseteq\operatorname{dom}(p)$

• There is a finite subset of $\omega$, $w_{p}$, such that $w_{p}=\operatorname{dom}(p)-\bigcup_{j

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

This satisfies ccc. To see this, consider any uncountable sequence $S=\langle p_{\alpha}\rangle_{\alpha<\omega_{1}}$ 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:

• $D_{n}=\{p\in P\mid n\in\operatorname{dom}(p)\}$ for $n<\omega$. This is obviously dense since any $p$ not already in $D_{n}$ can be extended to one which is by adding $\langle n,1\rangle$

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

• For each $\alpha\notin S$, $n<\omega$, $D_{n,\alpha}=\{p\in P\mid m\geq n\wedge p(m)=0\}$ for some $m<\omega$. This is dense since if $p\notin D_{n,\alpha}$ then $\operatorname{dom}(p)\cap A_{\alpha}=A_{\alpha}\cap\left(w_{p}\cup\bigcup_{j}A% _{i_{j}}\right)$. 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 $m\leq x\in A_{\alpha}$. So $p\cup\{\langle x,0\rangle\}\in D_{n,\alpha}$.

By $MA_{\kappa}$, given any set of $\kappa$ dense subsets of $P$, there is a generic $G$ which intersects all of them. There are a total of $\aleph_{0}+|S|+(\kappa-|S|)\cdot\aleph_{0}=\kappa$ 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 $n<\omega$, $G\cap D_{n}$ is non-empty, $n\in\operatorname{dom}(g)$, 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 $n<\omega$, $\alpha\notin S$, the set of $n\in A_{\alpha}$ such that $g(n)=0$ is unbounded, and hence infinite. So $g$ is as promised, and $2^{\kappa}=2^{\aleph_{0}}$.

Title Martin’s axiom and the continuum hypothesis MartinsAxiomAndTheContinuumHypothesis 2013-03-22 12:55:05 2013-03-22 12:55:05 Henry (455) Henry (455) 4 Henry (455) Result msc 03E50