I'm stuck on the following exercise and would appreciate hints on how to proceed. I haven't worked much with MA before.
Assume MA. Given a countable independent family $\mathcal{A} \subseteq [\omega]^\omega$, show that there exists an uncountable independent family $\mathcal{B} \subseteq [\omega]^\omega$ such that $\mathcal{B}$ contains $\mathcal{A}$.
My basic understanding is that I first have to find a suitable ccc forcing notion $\mathbb{P}$, then certain dense subsets of it to approximate the desired object to solve the question. Since one can formulate the definition of an independent family in terms of finite partial functions, I let $\mathbb{P}$ be given as $Fn(I,J)$, the set of all the finite partial functions $I \to J$. This forcing notion has the ccc iff $I \neq \emptyset$ or $J$ is countable. (I was thinking about letting $I = \mathcal{A}$ and $J = \{0,1\}$.)
Next, I would like to find certain subsets which are dense in $\mathbb{P}$ so that by MA, there is a filter $G \subseteq \mathbb{P}$ that meets them. The aim is to make $\bigcup G$ have certain properties, if $G$ meets those sets.
I know how to specify dense sets so that $\bigcup G$ is a welldefined onto function $I \to J$. But how can I find dense sets in order to "preserve" independence? Also, I fail to see the right connection between $\bigcup G$ and the desired $\mathcal{B}$. Maybe my approach does not make sense? Any hints are welcome!