The Deep Structure of the Real line For the sake of brevity, throughout this post I will identify real numbers with subsets of $\mathbb{N}$. The question that I want to ask here is more heuristic than definite; I want to understand the deeper intuition behind the fact that the Continuum Hypothesis is independent of the ZFC axioms. I am familiar with Godel's and Cohen's proof showing the independence of CH, and now I am trying to figure out the deeper heuristic behind them. 
If we compare another famous independence problem, the independence of Euclid's parallel postulate we notice (with great hindsight) that the reason why this postulate is independent is because the concept of 'line' has not been sufficiently well defined. I.e the concept of a line depends on the ambient space you are working in; on a sphere, a line has very different properties than on a plane.
When we transfer this intuition to the Theory of Sets, we see that there is a set theoretic concept which similarly lacks a clear definition, namely that of 'subset'. Take $P(\mathbb{N}) = \mathbb{R}$ for example. All the subsets we have ever encountered are 'definable' in the sense that they can be recursively enumerated by a finite algorithm; e.g.$$\{2,4,6,8,...\}$$ $$\{2,3,5,7,...\}$$ $$\{1,4,9,16,...\}$$
etc... and since an algorithm is just a finite string of symbols, there are countably many 'definable' reals. But we know that $\mathbb{R}$ is uncountable. Therefore there must be real numbers hidden 'deep' within $\mathbb{R}$, which we can call the deep structure of $\mathbb{R}$. Now if my intuition is correct, it is precisely the fact that these 'deep' reals cannot be defined in ZFC that the exact number of them (i.e. $|\mathbb{R}|$) is undecided by the rest of ZFC. I would like to make this notion more definite.
So I ask: Is this the right way to understand the deeper reason behind the independence of CH? Are there any expository articles been written on this subject that I can access? (I am not really interested in things like  PFA $\implies 2^{{\aleph _{0}}}=\aleph _{2}$ right now). Can anyone here explain intuitively why the undefinability of most real numbers allows for us to add arbitrarily large numbers of reals to $\mathbb{R}$ (I.e. in Cohen real forcing). Many thanks.
 A: I think that what you're missing here—indeed, what many people miss—is that the Continuum Hypothesis is not a statement about the reals. It is a statement about the power set of the reals.
To see why, note that you can always force CH to hold without adding any reals to your model.1 But you cannot change the truth value of CH without adding new subsets of $\Bbb R$. If we translate this to the von Neumann hierarchy, CH is not a statement about $V_{\omega+1}$ (which is practically $\Bbb R$), but rather about $V_{\omega+2}$.
Now, let's talk about the intuition here. Let's fix a model of $\sf ZFC$, call it $V$. The von Neumann hierarchy tells you that the universe is generated by simply at each step taking all the subsets available to you.
The constructible hierarchy, on the other hand, lets you limit what you're adding to your universe, by letting you only add things that you can define. This lets you have a much finer structure to the universe, it lets you refine its construction, and thus provides you with a way to prove things you cannot prove about an arbitrary model of $\sf ZFC$.
This is not about undefinability of real numbers. This is not about undefinability at all. This is about sets of real numbers. And the reason that we cannot really prove anything about CH from $\sf ZFC$ itself is that the power set operation is "too wild" for $\sf ZFC$ to control with only first-order logic statements.2
So whereas $\Bbb N$ is something that $\sf ZFC$ has a firm and tight grasp over, its power set is a much looser object, and its power set is well outside the grasp of $\sf ZFC$. 
This may or may not answer your question. And you might read this and exasperate in annoyance, that this is not what you signed up for, you're not even supposed to be working today. But that's set theory for you, indeed that's mathematics for you. Some ideas cannot be communicated "intuitively", unless you already have the right intuition to begin with. And while this is unfortunate, I can only comfort you by reminding you that beer is still a tangible object that not even $\sf ZFC$ can ruin for you.

Footnotes.


*

*With the usual caveat that we're talking about countable models. But that's not the important part here.

*You could argue that second-order $\sf ZFC$ is better here. But this is  bit trickier. Any model of second-order $\sf ZFC$ is of the form $V_\kappa$ for some inaccessible $\kappa$, so it must agree with $V$ on the truth value of CH. This means that we delegate the question about CH to the meta-theory, to $V$, and then the question reverts to its original form.
