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 $C H$ implies $M A$ .

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 $\{0,1\}$
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There exist $i_{1},\ldots,i_{n}\in S$ such that for each $j<n$ , $A_{i_{j}}\subseteq\operatorname{dom}(p)$
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There is a finite subset of $\omega$ , $w_{p}$ , such that $w_{p}=\operatorname{dom}(p)-\bigcup_{j<n}A_{i_{j}}$
•

For each $j<n$ , $p(a)=0$ 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
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

Martin’s axiom and the continuum hypothesis

MAℵ0<math id="S0.SSx1.m1" class="ltx_Math" alttext="MA_{\aleph_{0}}" display="inline"><mrow><mi>M</mi><mo>⁢</mo><msub><mi>A</mi><msub><mi mathvariant="normal">ℵ</mi><mn>0</mn></msub></msub></mrow></math> always holds

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