Cardinals, bijections, and a general inquiry

I'm trying to solve the following problem from Schimmerling's "A Course on Set Theory."

(Problem) Prove that there exists a family $\mathcal G\subseteq\mathcal P(\omega)$ such that $|\mathcal G|=2^{\aleph_0}$ and that for all $A,B\in\mathcal G$, if $A\neq B$ then $A\cap B$ is finite.

This problem has previous parts. First, it asks to show that the ordinal $^{<\omega}2=\bigcup_{n<\omega}{^{n}2}$ is countable. For this, I argued that $^{<\omega}2$ is a subset of $\omega$ (indeed, $^{<\omega}2\subseteq\bigcup_{n<\omega}n=:\omega)$). Second, it asks to show that the set

$$\mathcal F=\{\{x\upharpoonright n:n<\omega\}\mid x\in~^\omega2\}$$

has cardinality $2^{\aleph_0}$. For this, I constructed a bijection between the elements of $\mathcal F$ and the elements of $2^{\aleph_0}$. Specifically, for every function from the set of natural numbers to $\{0,1\}$, there is a corresponding function of the form $\{x\upharpoonright n:n<\omega\}$. (The latter set sort of describes the function recursively; so I'm thinking that $\mathcal F$ is in fact the set of all possible functions, recursively defined, from $\omega$ to $\{0,1\}$.)

The issue I'm facing now is to apply the previous results to prove the main problem. While a hint was given to first show that $\mathcal F\subseteq\mathcal P(^{<\omega}2)$, I can't derive it. (My attempts to biject $\mathcal F$ with a subset of $\mathcal P(^{<\omega}2)$ have failed.)

Assuming I could derive it, I also cannot seem to see its usefulness. If $\mathcal F$ is supposed to be the $\mathcal G$ in the problem, then it does not seem apparent that for all $A,B\in\mathcal F$ we yield a finite intersection $A\cap B$. (E.g., let $A$ and $B$ -- two functions from $\omega$ to $\{0,1\}$-- agree ultimately.)

I suspect that there is something mistaken in my interpretation of one (or more) of the sets above. Indeed, I would also like to ask if there are any general methods or tricks I could use to understand/interpret sets that are defined over many "levels" of quantifiers and predicates, such as $\mathcal F$ above. Also, are there useful ways to understand cardinals/ordinals beyond their definitions that lead to quicker and less confusing solutions?

• One thing that may be confusing you is that $^{\mathord{<}ω}2$, which is defined as $\bigcup_{n<\omega}{^n2}$, is not an ordinal. It the set of functions from natural numbers to 2. – Trevor Wilson Nov 27 '13 at 16:45

HINT: Think of $^{<\omega}2$ as a tree. Branches are exactly those of the form $\{x\upharpoonright n\mid n\in\omega\}$ for some $x\in{}^2\omega$. Show that two different branches must have a finite intersection. Then by the countablility of the tree the result transfers to $\omega$ itself.
• It is important because I want to tell the difference between finitely many Woodin cardinals and $\omega$ many Woodin cardinals :-) – Trevor Wilson Nov 27 '13 at 17:07