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:

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

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There exist ${i}_{1},\mathrm{\dots},{i}_{n}\in S$ such that for each $$, ${A}_{{i}_{j}}\subseteq \mathrm{dom}(p)$

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There is a finite subset of $\omega $, ${w}_{p}$, such that $$

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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:

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${D}_{n}=\{p\in P\mid n\in \mathrm{dom}(p)\}$ for $$. This is obviously dense since any $p$ not already in ${D}_{n}$ can be extended to one which is by adding $\u27e8n,1\u27e9$

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${D}_{\alpha}=\{p\in P\mid \mathrm{dom}(p)\supseteq {A}_{\alpha}\}$ for $\alpha \in S$. This is dense since if $p\notin {D}_{\alpha}$ then $p\cup \{\u27e8a,1\u27e9\mid a\in {A}_{\alpha}\setminus \mathrm{dom}(p)\}$ is.

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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 $p\cup \{\u27e8x,0\u27e9\}\in {D}_{n,\alpha}$.
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 $$, $G\cap {D}_{n}$ is nonempty, $n\in \mathrm{dom}(g)$, so $g$ is a total function. Since $G\cap {D}_{\alpha}$ for $\alpha \in S$ is nonempty, there is some element of $G$ whose domain contains all of ${A}_{\alpha}$ and is $0$ on a finite number of them, hence $g(a)=0$ for a finite number of $a\in {A}_{\alpha}$. Finally, since $G\cap {D}_{n,\alpha}$ for each $$, $\alpha \notin S$, 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  20130322 12:55:05 
Last modified on  20130322 12:55:05 
Owner  Henry (455) 
Last modified by  Henry (455) 
Numerical id  4 
Author  Henry (455) 
Entry type  Result 
Classification  msc 03E50 